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BANACH J. MATH. ANAL. (TO APPEAR)

TRIEBEL-LIZORKIN-TYPE SPACES WITH VARIABLEEXPONENTS

DACHUN YANG1, CIQIANG ZHUO1 AND WEN YUAN1∗

Abstract. In this article, the authors first introduce the Triebel-Lizorkin-

type space Fs(·),φp(·),q(·)(R

n) with variable exponents, and establish its ϕ-transform

characterization in the sense of Frazier and Jawerth, which further implies thatthis new scale of function spaces is well defined. The smooth molecular and the

smooth atomic characterizations of Fs(·),φp(·),q(·)(R

n) are also obtained, which are

used to prove a trace theorem of Fs(·),φp(·),q(·)(R

n). The authors also characterize

the space Fs(·),φp(·),q(·)(R

n) via Peetre maximal functions.

1. Introduction

Between 1960’s and 1970’s, the Besov space Bsp,q(R

n) and the Triebel-Lizorkinspace F s

p,q(Rn) were introduced and investigated accompanying with the devel-

opment of the theory of function spaces (see, for example, [66]). These spacesform a very general unifying scale of many well-known classical concrete functionspaces such as Lebesgue spaces, Holder-Zygmund spaces, Sobolev spaces, Bessel-potential spaces, Hardy spaces and BMO, which have their own history. A com-prehensive treatment of these function spaces and their history can be foundedin Triebel’s monographes [66, 67, 68, 69]. Recently, to clarify the relations amongBesov spaces, Triebel-Lizorkin spaces and Q spaces (see [16, 24]), Besov-typespaces Bs,τ

p,q (Rn) and Triebel-Lizorkin-type spaces F s,τ

p,q (Rn) and their homoge-

neous counterparts for all admissible parameters were introduced and studied in[76, 77, 80]. Moreover, the Besov-type and the Triebel-Lizorkin-type spaces, in-cluding some of their special cases related to Q spaces, have been used to studythe existence and the regularity of solutions of some partial differential equationssuch as (fractional) Navier-Stokes equations; see, for example, [43, 44, 45, 70, 81].For more properties of these spaces, we refer the reader to [63, 64, 78, 79].

On the other hand, in recent years, there has been a growing interesting ingeneralizing classical spaces such as Lebesgue and Sobolev spaces to cases with

Date: Received: Sep. 21, 2014; Accepted: Jan. 4, 2015.∗ Corresponding author.2010 Mathematics Subject Classification. Primary 46E35; Secondary 42B25, 42B35.Key words and phrases. Triebel-Lizorkin space, variable exponent, molecule, atom, trace.

1

http://arxiv.org/abs/1503.04510v1

2 D. YANG, C. ZHUO, W. YUAN

either variable integrability or variable smoothness (see [13, 19]), which are ob-viously not covered by any function space with invariable exponents. Spacesof variable integrability can be traced back to Birnbaum-Orlicz [10], Orlicz [58]and Nakano [54, 55]. In particular, the definition of so-called Musielak-Orliczspaces was clearly written by Nakano in [54, Section 89], while it seems thatOrlicz was mainly interested in the completeness of function spaces. But themodern development was started with the article [37] of Kovacik and Rakosnıkin 1991 and widely used in the study of harmonic analysis as well as partialdifferential equations; see, for example, [6, 12, 13, 14, 17, 18, 19, 21, 32, 48].The motivation to study such function spaces also comes from applications tofluid dynamic, image processing and the calculus of variation; see, for example,[1, 2, 3, 11, 21, 25, 59, 60].

To complete the theory of the variable exponent Lebesgue and Sobolev spaces,Almeida and Samko [5] and Gurka et al. [30] introduced and investigated variableexponent Bessel potential spaces Lα,p(·) with variable integrability index p(·).Later, Xu [73, 74, 75] studied Besov spaces Bs

p(·),q(Rn) and Triebel-Lizorkin spaces

F sp(·),q(R

n) with the variable exponent p(·) but invariable exponents q and s. Alonga different line of study, when Leopold [38, 39, 40, 41] and Leopold and Schrohe[42] studied pseudo-differential operators with symbols of the type

(1 + |ξ|2)s(x)/2,they defined and investigated related Besov spaces with variable smoothness,

Bs(·)p,p (Rn). Function spaces of variable smoothness including Besov space B

s(·)p,q (Rn)

and Triebel-Lizorkin space Fs(·)p,q (Rn) have been studied by Besov [7, 8, 9], which

was a generalization of Leopold’s work. Another interesting research direction offunction spaces with variable integrability is the theory of Hardy spaces Hp(·)(Rn)with variable exponents as well as local Hardy spaces hp(·)(Rn), which was intro-duced and investigated by Nakai and Sawano [53] and they proved that

hp(·)(Rn) = F 0p(·),2(R

n).

Independently, Cruz-Uribe and Wang in [15] also investigated the variable expo-nent Hardy space with some weaker conditions than those used in [53].

As we can see from the trace and the embedding theorems of the classicalfunction spaces, the smoothness and the integrability often interact each other;see, for example, [80, Theorem 6.8 and Corollary 2.2]. As was pointed out in [4,p. 1629] and [20, p. 1733], the unifications of the trace and the Sobolev embeddingdo not occur on function spaces with only one index variable. For example, thetrace space of the variable exponent Sobolev space W k,p(·) is no longer a spaceof the same type (see [19]), since they involve an interaction between integrabil-ity and smoothness. As one of motivations, to tackle this problem, Alexandreand Hasto [20] introduced and investigated Triebel-Lizorkin spaces with variable

smoothness and integrability Fs(·)p(·),q(·)(R

n) with s(·) ≥ 0, and showed that these

spaces behaved nicely with respect to the trace operator. Subsequently, Vybıral[72] established Sobolev and Jawerth embeddings of these spaces and, moreover,

Kempka [34] characterized Fs(·)p(·),q(·)(R

n) by local means, and Kempka and Vybıral

TRIEBEL-LIZORKIN-TYPE SPACES 3

[36] obtained the equivalent characterization via ball means of differences. The

main difficulty of studying Fs(·)p(·),q(·)(R

n) is the absence of the vector-valued in-

equality for the boundedness on Lp(·)(ℓq(·)(Rn)) of the Hardy-Littlewood maximalfunction, which, in the classical case with p, q, s being constant exponents, isa very important tool in studying the space F s

p,q(Rn). The vector-valued con-

volution inequality developed in [20, Theorem 3.2] (see also Lemma 2.9 below)supplies a well remedy for this absence.

Vybıral [72] and Kempka [34] also studied the Besov space Bs(·)p(·),q(R

n) with the

only index q being a constant, which is a quite natural case, since the norm in theBesov space is usually defined via the iterated space ℓq(Lp(Rn)). Furthermore,

Almeida and Hasto [4] introduced and investigated the Besov space Bs(·)p(·),q(·)(R

n)

with all three variable exponents, which makes a further step in completing theunification process of function spaces with variable smoothness and integrability.

The atomic characterization of Bs(·)p(·),q(·)(R

n) was established by Drihem [22] and

some equivalent characterizations via local means and ball means of differenceswere also obtained by Kempka and Vybral [36]. Moreover, Noi and Sawano [57] in-vestigated the complex interpolation of Besov spaces and Triebel-Lizorkin spaceswith variable exponents (see also [79] for the complex interpolation of Besov-type spaces and Triebel-Lizorkin-type spaces but with invariable exponents) and,in [56], Noi studied the trace and the extension operators for Beosv spaces andTriebel-Lizorkin spaces with variable exponents. Very recently, Izuki et al. [33]gave out an elementary introduction to function spaces with variable exponentsand a survey of related function spaces.

More generally, Kempka [35] introduced and studied 2-microlocal Besov andTriebel-Lizorkin spaces with variable integrability and gave out characterizationsby decompositions in atoms, molecules and wavelets, which cover the usual Besovand Triebel-Lizorkin spaces as well as spaces of variable smoothness and integra-

bility and also include the space Fs(·)p(·),q(·)(R

n) without the restriction s(·) ≥ 0. The

trace of 2-microlocal Besov and Triebel-Lizorkin spaces with variable exponentswas studied by Moura et al. [49], as well as Goncalves et al. [29]. Moreover, Ho[31] investigated the variable Triebel-Lizorkin-Morrey space, which is an exten-sion of Triebel-Lizorkin-Morrey spaces in [61, 62] and also generalizes the function

space Fs(·)p(·),q(·)(R

n) in [20].

Here, we should point out that, different from the classical case with exponents

being constants, the definition of Bs(·)p(·),q(·)(R

n) is more complicated than that of

Fs(·)p(·),q(·)(R

n). The main reason is that the mixed Lebesgue-sequence space

ℓq(·)(Lp(·)(Rn))

(see [4, Definition 3.1]) involved in the definition of Bs(·)p(·),q(·)(R

n), as was pointed

out by Almeida and Hasto in [4, Remark 4.2], does not enjoy one key featureof iterated function spaces, namely, inheritance of properties from constituentspaces. To limit the length of this article, we leave the study of Besov-typespaces with all variable exponents in a furthercoming article.


The purpose of this article is to introduce and study a more generalized scale offunction spaces, based on the Triebel-Lizorkin-type space F s,τ

p,q (Rn), with variable

exponent of smoothness, s(·), variable exponents of integrability, p(·) and q(·),and a set function φ, denoted by F

s(·),φp(·),q(·)(R

n). These spaces generalize classical

Triebel-Lizorkin-type spaces and Triebel-Lizorkin spaces with variable smooth-ness and integrability. Molecular and atomic characterizations, Peetre maximalfunction characterizations of these spaces are also established in this article. Asapplications, we show a trace theorem of Triebel-Lizorkin-type spaces with vari-able exponents and give out some equivalent quasi-norms under some restrictionsof the set function φ.

We begin with some basic notation. In what follows, for a measurable functionp(·) : Rn → (0,∞) and a measurable set E of Rn, let

p−(E) := ess infx∈E

p(x) and p+(E) := ess supx∈E

p(x).

For notational simplicity, we let p− := p−(Rn) and p+ := p+(R

n). Denote byP(Rn) the collection of all measurable functions p(·) : Rn → (0,∞) satisfying0 < p− ≤ p+ <∞.

For p(·) ∈ P(Rn) and a measurable set E ⊂ Rn, the space Lp(·)(E) is definedto be the set of all measurable functions f such that

‖f‖Lp(·)(E) := inf

λ ∈ (0,∞) :

∫

E

[ |f(x)|λ

]p(x)dx ≤ 1

<∞.

For r ∈ (0,∞), denote by Lrloc(Rn) the set of all r-locally integrable functions on

Rn. Denote by L∞(Rn) the set of all measurable functions f such that

‖f‖L∞(Rn) := ess supy∈Rn

|f(y)| <∞.

Remark 1.1. Let p(·) ∈ P(Rn).(i) It was presented in [53, p. 3671] (see also [13, Theorem 2.17]) that, for all

λ ∈ C,

‖λf‖Lp(·)(Rn) = |λ|‖f‖Lp(·)(Rn)

and, if r ∈ (0,minp−, 1], then, for all f, g ∈ Lp(·)(Rn),

‖f + g‖rLp(·)(Rn) ≤ ‖f‖rLp(·)(Rn) + ‖g‖rLp(·)(Rn).

(ii) If∫

Rn

[ |f(x)|δ

]p(x)dx ≤ C

for some δ ∈ (0,∞) and some positive constant C independent of δ, then it iseasy to see that ‖f‖Lp(·)(Rn) ≤ Cδ, where C is a positive constant independent of

δ, but depending on p− (or p+) and C.(iii) Let p(·) ∈ P(Rn) satisfy 1 < p− ≤ p+ <∞. Define the conjugate exponent

P (·) of p(·) by setting, for all x ∈ Rn, P (x) := p(x)p(x)−1

. It was proved, in [13,


Theorem 2.6], that, if f ∈ Lp(·)(Rn) and g ∈ LP (·)(Rn), then fg ∈ L1(Rn) and∫

Rn

|f(x)g(x)| dx ≤ C‖f‖Lp(·)(Rn)‖g‖LP (·)(Rn),

where C is a positive constant depending on p− or p+, but independent of f andg.

(iv) Obviously, the space Lp(·)(Rn) has the lattice property, namely, if |f | ≤ |g|,then

‖f‖Lp(·)(Rn) ≤ ‖g‖Lp(·)(Rn).

Recall that a measurable function g ∈ P(Rn) is said to satisfy the locally log-

Holder continuous condition, denoted by g ∈ C logloc (R

n), if there exists a positiveconstant Clog(g) such that, for all x, y ∈ Rn,

|g(x)− g(y)| ≤ Clog(g)

log(e + 1/|x− y|) , (1.1)

and g is said to satisfy the globally log-Holder continuous condition, denoted byg ∈ C log(Rn), if g ∈ C log

loc (Rn) and there exist positive constants C∞ and g∞ such

that, for all x ∈ Rn,

|g(x)− g∞| ≤ C∞log(e+ |x|) .

Remark 1.2. (i) Let g ∈ C log(Rn). Then g∞ = lim|x|→∞ p(x).(ii) Let g ∈ P(Rn). Then g ∈ C log(Rn) if and only if 1/g ∈ C log(Rn).

For all x ∈ Rn and r ∈ (0,∞), denoted by Q(x, r) the cube centered at x withside-length r, whose sides parallel axes of coordinates. Let φ : Rn × [0,∞) →(0,∞) be a measurable function. In this article, we always suppose that φ satisfiesthe following two conditions:

(S1) there exist positive constants c1 and c1 such that, for all x ∈ Rn andr ∈ (0,∞),

(c1)−1 ≤ φ(x, r)

φ(x, 2r)≤ c1;

(S2) there exists a positive constant c2 such that, for all x, y ∈ Rn and r ∈(0,∞) with |x− y| ≤ r,

(c2)−1 ≤ φ(x, r)

φ(y, r)≤ c2.

In what follows, for all cubes Q := Q(x, r) with x ∈ Rn and r ∈ (0,∞), let

φ(Q) := φ(Q(x, r)) := φ(x, r).

Remark 1.3. (i) We point out that the conditions (S1) and (S2) of φ are, respec-tively, called doubling condition and compatibility condition, which have beenused by Nakai [50, 51] and Nakai and Sawano [53] when they studied generalizedCampanato spaces.

(ii) Let φ(Q) := |Q|τ with τ ∈ [0,∞) for all cubes Q. Then, obviously, φsatisfies the conditions (S1) and (S2).


(iii) Let p(·) ∈ C log(Rn). The set function φ, defined by setting, for all cubesQ,

φ(Q) :=‖χQ‖Lp(·)(Rn)

|Q| ,

which is just [53, Example 6.4], satisfies the conditions (S1) and (S2).(iv) Let φ be a nondecreasing set function, namely, there exists a positive

constant C such that, for all cubes Q1 ⊂ Q2, φ(Q1) ≤ Cφ(Q2). If φ satisfies thecondition (S1), then φ also satisfies the condition (S2). Indeed, for all x, y ∈ Rn

and r ∈ (0,∞) with |x − y| ≤ r, it is easy to see that Q(x, r) ⊂ Q(y, 2r) andQ(y, r) ⊂ Q(x, 2r) and, by the condition (S1), we see that

1 .φ(x, r)

φ(x, 2r).φ(x, r)

φ(y, r).φ(y, 2r)

φ(y, r). 1

with the implicit positive constants independent of x, y and r. Thus, φ satisfiesthe condition (S2).

(v) Let φ(Q) :=∫Qw(x) dx for all cubes Q, where w is a classical Muckenhoupt

Ap(Rn)-weight with p ∈ [1,∞]. It is well known that each Muckenhoupt Ap(R

n)-weight is doubling, thus, by (iv), we conclude that φ satisfies the conditions (S1)and (S2). For the definition and properties of Muckenhoupt Ap(R

n)-weights, werefer the reader to [65].

Let S(Rn) be the space of all Schwartz functions on Rn and S ′(Rn) its topologi-cal dual space. We say a pair (ϕ,Φ) of functions to be admissible if ϕ, Φ ∈ S(Rn)satisfy

supp ϕ ⊂ξ ∈ Rn :

1

2≤ |ξ| ≤ 2

and |ϕ(ξ)| ≥ c > 0 when

3

5≤ |ξ| ≤ 5

3(1.2)

and

supp Φ ⊂ ξ ∈ Rn : |ξ| ≤ 2 and |Φ(ξ)| ≥ c > 0 when |ξ| ≤ 5

3, (1.3)

where f(ξ) :=∫Rn f(x)e

−ix·ξ dx for all ξ ∈ Rn and f ∈ L1(Rn), and c is a positiveconstant independent of ξ ∈ Rn. Throughout the article, for all ϕ ∈ S(Rn),

j ∈ N := 1, 2, . . . and x ∈ Rn, we put ϕj(x) := 2jnϕ(2jx) and ϕ(x) := ϕ(−x).For j ∈ Z and k ∈ Zn, denote by Qjk the dyadic cube 2−j([0, 1)n + k), xQjk

:=

2−jk its lower left corner and ℓ(Qjk) its side length. Let

Q := Qjk : j ∈ Z, k ∈ Zn, Q∗ := Q ∈ Q : ℓ(Q) ≤ 1and jQ := − log2 ℓ(Q) for all Q ∈ Q.

Now we introduce Triebel-Lizorkin-type spaces with variable exponents.

Definition 1.4. Let (ϕ,Φ) be a pair of admissible functions on Rn. Let p, q ∈P(Rn) satisfy

0 < p− ≤ p+ <∞, 0 < q− ≤ q+ <∞and 1

p, 1q∈ C log(Rn), s ∈ C log

loc (Rn) ∩ L∞(Rn) and φ be a set function satisfying

the conditions (S1) and (S2). Then the Triebel-Lizorkin-type space with variable


exponents, Fs(·),φp(·),q(·)(R

n), is defined to be the set of all f ∈ S ′(Rn) such that

‖f‖F

s(·),φp(·),q(·)

(Rn):= sup

P∈Q

1

φ(P )

∥∥∥∥∥∥∥

∞∑

j=maxjP ,0

[2js(·)|ϕj ∗ f(·)|

]q(·)

1q(·)

∥∥∥∥∥∥∥Lp(·)(P )

<∞,

where, when j = 0, ϕ0 is replaced by Φ, and the supremum is taken over alldyadic cubes P in Rn.

Remark 1.5. Let p(·), q(·), s(·) be as in Definition 1.4.(i) When φ(Q) := 1 for all cubes Q, then

Fs(·),φp(·),q(·)(R

n) = Fs(·)p(·),q(·)(R

n),

where Fs(·)p(·),q(·)(R

n) denotes the Triebel-Lizorkin space with variable smoothness

and integrability which is introduced and investigated in [20]. We point out that

Diening et al. [20] studied the space Fs(·)p(·),q(·)(R

n) under an additional assumption

that s is nonnegative, which is generalized to the case that s : Rn → R ands ∈ C log

loc (Rn) ∩ L∞(Rn) by Kempka in [35].

(ii) When p, q, s are constant exponents and φ(Q) := |Q|τ with τ ∈ [0,∞) forall cubes Q, then

Fs(·),φp(·),q(·)(R

n) = F s,τp,q (R

n),

where F s,τp,q (R

n) denotes the Triebel-Lizorkin-type space which was introduced andstudied in [80].

(iii) When q, s are constant exponents and φ(Q) := |Q|τ with τ ∈ [0,∞) forall cubes Q, then

Fs(·),φp(·),q(·)(R

n) = F s,τp(·),q(R

n),

which was investigated in [47].(iv) The condition, 0 < p− ≤ p+ < ∞, is quite natural, since there also

exists the restriction p < ∞ in the case of constant exponents. The assumption,0 < q− ≤ q+ < ∞, is different from the case of constant exponents whereq = ∞ is included. This restriction comes from the application of the convolutioninequality in [20, Theorem 3.2] (see also Lemma 2.9 below), when proving that

the space Fs(·),φp(·),q(·)(R

n) is independent of the choice of admissible function pairs

(ϕ,Φ). Observe that, even when φ(Q) = 1 for all cubes Q, this restriction isnecessary; see [36, p. 857].

This article is organized as follows.

Section 2 is devoted to showing that the space Fs(·),φp(·),q(·)(R

n) is independent of

the choice of admissible function pairs (ϕ,Φ), which is a consequence of the ϕ-

transform characterization of Fs(·),φp(·),q(·)(R

n) in the sense of Frazier and Jawerth

(see Corollary 2.4 below). Different from the method used in the case of con-stant exponents, in the proof of the boundedness of the ϕ-transform Sϕ from

Fs(·),φp(·),q(·)(R

n) to fs(·),φp(·),q(·)(R

n) (the sequence space corresponding to the function

space Fs(·),φp(·),q(·)(R

n)), we make full use of the so-called r-trick lemma (namely, [20,


Lemma A.6]) and the vector-valued convolution inequality (namely, [20, Theorem3.2]; see also Lemma 2.9 below). We point out that the vector-valued convolutioninequality also plays an essential role throughout the remainder of this article.

In Section 3, we establish equivalent characterizations of Fs(·),φp(·),q(·)(R

n) in term of

molecules, atoms (see Theorem 3.8 below) or Peetre maximal functions (see The-orem 3.11 below). To prove Theorem 3.8, we borrow some ideas from the proof of[23, Theorem 3.12] which gives the atomic characterization of the Besov-type andthe Triebel-Lizorkin-type spaces, and the proof of [35, Theorem 3.13] which givesthe molecular characterization of 2-microlocal Besov and Triebel-Lizorkin spaceswith variable integrability. The Sobolev embedding (see Proposition 3.1 below)plays a key role in the proof of Theorem 3.8, which may be of independent inter-est. The proof of Theorem 3.11 is similar to that of [46, Theorem 3.2] (see also [71,Theorem 2.6]) and strongly depends on the vector-valued convolution inequalityon Lp(·)(ℓq(·)(Rn)); see Lemma 2.9 below. As applications of Theorem 3.11, some

equivalent norms of Fs(·),φp(·),q(·)(R

n) are obtained (see Theorem 3.12 below), which

are further used to show that the spaces Fs(·),φp(·),q(·)(R

n) include the Morrey space

with variable exponents Mp(·)φ (Rn) as a special case; see Proposition 3.18 below.

At the end of Section 3, via some examples, we show that, in general, the scalesof Triebel-Lizorkin-type spaces with variable exponents and variable Triebel-Lizorkin-Morrey spaces (see [31]) do not cover each other (see Remark 3.15 below).Here we point out that Triebel-Lizorkin-type spaces with variable exponents inthis article cover the Triebel-Lizorkin-type spaces F s,τ

p,q (Rn) for all τ ∈ [0,∞), but

the variable Triebel-Lizorkin-Morrey space in [31] does only cover the Triebel-Lizorkin-type space F s,τ

p,q (Rn) for τ ∈ [0, 1/p).

In Section 4, as an application of the atomic characterization of Fs(·),φp(·),q(·)(R

n),

we mainly establish a trace theorem of Triebel-Lizorkin-type spaces with variableexponents (see Theorem 4.1 below). In the case that φ is as in Remark 1.5(i),the corresponding result of Theorem 4.1 was obtained in [20, Theorem 3.13] witha certain weaker condition (see Remark 4.2(ii) below), however, the convergence

of the trace of f ∈ Fs(·)p(·),q(·)(R

n) was not given out exactly in [20, p. 1760]. In

Section 4, we first show that the trace operator is well defined on the space

Fs(·),φp(·),q(·)(R

n) (see Lemma 4.3 below), with a certain restriction on p and s, by an

argument similar to that used in the proof of Theorem 3.8(i). Indeed, in Lemma

4.3 below, we prove that the trace of f ∈ Fs(·),φp(·),q(·)(R

n) converges in S ′(Rn−1).

Then, similar to the proof of [20, Theorem 3.13], we show that the trace space of

Fs(·),φp(·),q(·)(R

n) is independent of the n-th coordinate of variable exponents p(·) ands(·), and complete the proof by an argument similar to that used in the proof of[80, Theorem 6.8].

Finally, we make some conventions on notation. Throughout this article, wedenote by C a positive constant which is independent of the main parameters,but may vary from line to line. The symbols A . B means A ≤ CB. If A . Band B . A, then we write A ∼ B. For all a, b ∈ R, let

a ∨ b := maxa, b.


If E is a subset of Rn, we denote by χE its characteristic function. For all cubesQ, we use cQ to denote its the center. For all k := (k1, . . . , kn) ∈ Zn, let

|k| := |k1|+ · · ·+ |kn|.Let N := 1, 2, . . . and Z+ := 0 ∪ N.

2. The ϕ-transform characterization

In this section, we first introduce the sequence space fs(·),φp(·),q(·)(R

n) corresponding

to the space Fs(·),φp(·),q(·)(R

n) and then establish their ϕ-transform characterization in

the sense of Frazier and Jawerth [27]. As a consequence of the ϕ-transform

characterization, we conclude that the space Fs(·),φp(·),q(·)(R

n) is independent of the

choice of admissible function pairs (ϕ,Φ).

Definition 2.1. Let p(·), s(·) and φ be as in Definition 1.4 and q(·) as either inDefinition 1.4 or q(·) ≡ ∞. Then the sequence space f

s(·),φp(·),q(·)(R

n) is defined to be

the set of all sequences t := tQQ∈Q∗ ⊂ C such that

‖t‖fs(·),φp(·),q(·)

(Rn):= sup

P∈Q

1

φ(P )

∥∥∥∥∥∥

∑

Q⊂P,Q∈Q∗

[|Q|−[ s(·)

n+ 1

2]|tQ|χQ

]q(·) 1

q(·)

∥∥∥∥∥∥Lp(·)(P )

<∞

with the usual modification made when q(·) ≡ ∞, where the supremum is takenover all dyadic cubes P in Rn.

Remark 2.2. (i) It is easy to see that fs(·),φp(·),q(·)(R

n) is a quasi-Banach lattice, namely,

for all t(1) := t(1)Q Q∈Q∗ ⊂ C and t(2) := t(2)Q Q∈Q∗ ⊂ C, if |t(1)Q | ≤ |t(2)Q | for allQ ∈ Q∗, then

‖t(1)‖fs(·),φp(·),q(·)

(Rn)≤ ‖t(2)‖

fs(·),φp(·),q(·)

(Rn).

(ii) Let D0(Rn) := Q ⊂ Rn : Q is a cube and ℓ(Q) = 2−j0 for some j0 ∈ Z.

Then it is easy to prove that the supremum in Definitions 1.4 and 2.1 can beequivalently taken over all cubes in D0(R

n), the details being omitted.

Let (ϕ,Φ) be a pair of admissible functions. Then (ϕ, Φ) is also a pair of ad-missible functions. Thus, by [27, pp. 130-131] or [28, Lemma (6.9)], we know thatthere exist Schwartz functions ψ and Ψ satisfying (1.2) and (1.3), respectively,such that, for all ξ ∈ Rn,

Φ(ξ)Ψ(ξ) +

∞∑

j=1

ϕ(2−jξ)ψ(2−jξ) = 1. (2.1)

Recall that the ϕ-transform Sϕ is defined to be the mapping taking each f ∈S ′(Rn) to the sequence Sϕ(f) := (Sϕf)QQ∈Q∗, where (Sϕf)Q := |Q|1/2Φ∗f(xQ)if ℓ(Q) = 1 and (Sϕf)Q := |Q|1/2ϕjQ ∗ f(xQ) if ℓ(Q) < 1; the inverse ϕ-transformTψ is defined to be the mapping taking a sequence t := tQQ∈Q∗ ⊂ C to

Tψt :=∑

Q∈Q∗, ℓ(Q)=1

tQΨQ +∑

Q∈Q∗, ℓ(Q)<1

tQψQ;


see, for example, [80, p. 31].

Now we state the following ϕ-transform characterization for Fs(·),φp(·),q(·)(R

n), which

is the main result of this section. For the corresponding result of Triebel-Lizorkin-type spaces, see [80, Theorem 2.1].

Theorem 2.3. Let p, q, s and φ be as in Definition 1.4 and ϕ, ψ, Φ and

Ψ be as in (2.1). Then the operators Sϕ : Fs(·),φp(·),q(·)(R

n) → fs(·),φp(·),q(·)(R

n) and

Tψ : fs(·),φp(·),q(·)(R

n) → Fs(·),φp(·),q(·)(R

n) are bounded. Furthermore, TψSϕ is the identity

on Fs(·),φp(·),q(·)(R

n).

We remark that Tψ is well defined for all t ∈ fs(·),φp(·),q(·)(R

n); see Lemma 2.5 below.

The proof of Theorem 2.3 is given later. From Theorem 2.3 and an argumentsimilar to that used in the proof of [27, Remark 2.6], we immediately deduce thefollowing conclusion, the details being omitted.

Corollary 2.4. With all the notation as in Definition 1.4, the space Fs(·),φp(·),q(·)(R

n)

is independent of the choice of the admissible function pairs (ϕ,Φ).

Now we start to show Theorem 2.3. First, we need the following property.

Lemma 2.5. Let p, q, s and φ be as in Definition 1.4. Then, for all t ∈fs(·),φp(·),q(·)(R

n),

Tψt :=∑

Q∈Q∗, ℓ(Q)=1

tQΨQ +∑

Q∈Q∗, ℓ(Q)<1

tQψQ

converges in S ′(Rn); moreover, Tψ : fs(·),φp(·),q(·)(R

n) → S ′(Rn) is continuous.

To prove Lemma 2.5, we need the following technical lemmas.

Lemma 2.6. Let φ be a set function satisfying the conditions (S1) and (S2).Then

(i) there exists a positive constant C such that, for any j ∈ Z+ and k ∈ Zn,

φ(Qjk) ≤ C2j log2 c1(|k|+ 1)log2(c1c1);

(ii) there exists a positive constant C such that, for all Q ∈ Q and l ∈ Zn,

φ(Q+ lℓ(Q))

φ(Q)≤ C(1 + |l|)log2(c1c1),

where c1 and c1 are as in the condition (S1).

Proof. We first prove (i). For any j ∈ Z+ and k ∈ Zn, let δk ∈ Z+ be suchthat 2δk ≤ n(|k| + 1) < 2δk+1 and cQjk

the center of the cube Qjk. Then, by the

conditions (S1) and (S2) of φ and the fact that |cQjk| ≤ n2−j(|k| + 1), we see

that

φ(Qjk) =φ(Q(cQjk, 2−j)) ≤ cδk+1

1 φ(Q(cQjk, 2−j+δk+1))

≤ c2cδk+11 φ(Q(0, 2−j+δk+1)) . cj+δk1 (c1)

δkφ(Q(0, 1))

. 2j log2 c1(|k|+ 1)log2(c1c1),


which completes the proof of (i).Next we show (ii). If |l| ≤ 1, namely, ℓ(Q)|l| ≤ ℓ(Q), then, by the condition

(S2) of φ, we find that

c−12 ≤ φ(Q+ lℓ(Q))

φ(Q)≤ c2. (2.2)

If |l| > 1, namely, ℓ(Q)|l| > ℓ(Q), then there exists a γl ∈ N such that 2γl ≤ |l| <2γl+1. Thus, ℓ(Q)2γl+1 > ℓ(Q)|l|. From the condition (S1) of φ, we deduce that

φ(Q+ lℓ(Q)) =φ(Q(cQ + lℓ(Q), ℓ(Q)))

≤ cγl+11 φ(Q(cQ + lℓ(Q), ℓ(Q)2γl+1))

and

φ(Q) = φ(Q(cQ, ℓ(Q))) ≥(

1

c1

)γl+1

φ(Q(cQ, ℓ(Q)2γl+1)),

which, combined with the condition (S2) of φ, implies that

φ(Q+ lℓ(Q))

φ(Q)≤ cγl+1

1 φ(Q(cQ + lℓ(Q), ℓ(Q)2γl+1))

(c1)−γl−1φ(Q(cQ, ℓ(Q)2γl+1))

≤ c2cγl+11 (c1)

γl+1 ∼ |l|log2(c1c1).This, together with (2.2), then finishes the proof of (ii) and hence Lemma 2.6.

Lemma 2.7. Let p(·) ∈ C log(Rn). Then there exists a positive constant C suchthat, for all dyadic cubes Qjk with j ∈ Z+ and k ∈ Zn,

1

C2− n

p−j(1 + |k|)n(

1p+

− 1p−

)≤‖χQjk‖Lp(·)(Rn)

≤C2− n

p+j(1 + |k|)n(

1p−

− 1p+

). (2.3)

Proof. Let Q00 be the dyadic cube Qjk with j = 0 and k = (0, . . . , 0) ∈ Zn. Forany j ∈ Z+ and k ∈ Zn, it is easy to see that Qjk ⊂ 2(1 + |k|)Q00. Then, from[82, Lemma 2.6], we deduce that

‖χQjk‖Lp(·)(Rn).

[ |Qjk||2(1 + |k|)Q00|

] 1p+

‖χ2(1+|k|)Q00‖Lp(·)(Rn)

.

[ |Qjk||2(1 + |k|)Q00|

] 1p+

[ |2(1 + |k|)Q00||Q00|

] 1p−

‖χQ00‖Lp(·)(Rn)

∼ 2− n

p+j(1 + |k|)n(

1p−

− 1p+

)

and, similarly,

‖χQjk‖Lp(·)(Rn) & 2

− np−

j(1 + |k|)n(

1p+

− 1p−

),

which completes the proof of (2.3) and hence Lemma 2.7.

In what follows, for h ∈ S(Rn) and M ∈ Z+, let

‖h‖SM(Rn) := sup|γ|≤M

supx∈Rn

|∂γh(x)|(1 + |x|)n+M+γ.


Proof of Lemma 2.5. To prove this lemma, it suffices to show that there exists

an M ∈ N such that, for all t ∈ fs(·),φp(·),q(·)(R

n) and h ∈ S(Rn),

|〈Tψt, h〉| . ‖t‖fs(·),φp(·),q(·)

(Rn)‖h‖SM (Rn).

Indeed, by Remark 1.1(iv), we see that, for any Q ∈ Q∗,

|tQ|= ‖tQχQ‖Lp(·)(Q)‖χQ‖−1Lp(·)(Q)

≤

∥∥∥∥∥∥∥∥∥

∑

Q⊂Q

Q∈Q∗

[|Q|− s(·)

n− 1

2 |tQ|χQ]q(·)

1q(·)

∥∥∥∥∥∥∥∥∥Lp(·)(Q)

‖χQ‖−1Lp(·)(Q)

|Q|s−n

+ 12

≤‖t‖fs(·),φp(·),q(·)

(Rn)‖χQ‖−1

Lp(·)(Q)φ(Q)|Q|

s−n

+ 12 ,

which implies that

|〈Tψt, h〉|≤∑

ℓ(Q)=1

|tQ||〈ΨQ, h〉|+∑

ℓ(Q)<1

|tQ||〈ψQ, h〉|

≤ ‖t‖fs(·),φp(·),q(·)

(Rn)

∑

ℓ(Q)=1

‖χQ‖−1Lp(·)(Q)

φ(Q)|〈ΨQ, h〉|

+∑

ℓ(Q)<1

|Q|s−n

+ 12‖χQ‖−1

Lp(·)(Q)φ(Q)|〈ψQ, h〉|

=: I1 + I2.

Let M ∈ N be such that

M > 2max

log2(c1c1) + n

(1

p−− 1

p+

)+n

2, log2 c1 +

n

p−− s− − 2n

.

Then, for I1, by Lemmas 2.6 and 2.7, we find that

I1≤‖f‖fs(·),φp(·),q(·)

(Rn)

∑

k∈Zn

‖χQ0k‖−1Lp(·)(Q0k)

φ(Q0k)|〈ΨQ0k, h〉|

. ‖f‖fs(·),φp(·),q(·)

(Rn)‖h‖SM (Rn)

∑

k∈Zn

(1 + |k|)n(1

p−− 1

p+)+log2(c1c1)−n−M

. ‖f‖fs(·),φp(·),q(·)

(Rn)‖h‖SM (Rn).

On the other hand, for I2, by [80, Lemma 2.4] and Lemmas 2.6 and 2.7, weconclude that

I2≤‖f‖fs(·),φp(·),q(·)

(Rn)‖h‖SM (Rn)

∞∑

j=1

∑

k∈Zn

2−j(s−+n

2−log2 c1− n

p−)2−jM

×(1 + |k|)n(1

p−− 1

p+)+log2(c1c1) 1

(1 + |2−jk|)n+M. ‖f‖

fs(·),φp(·),q(·)

(Rn)‖h‖SM (Rn),


which, together with the estimate of I1, implies that

|〈Tψt, h〉| . ‖t‖fs(·),φp(·),q(·)

(Rn)‖h‖SM (Rn).

Thus,

Tψt =∑

Q∈Q∗, ℓ(Q)=1

tQΨQ +∑

Q∈Q∗, ℓ(Q)<1

tQψQ

converges in S ′(Rn) and Tψ : fs(·),φp(·),q(·)(R

n) → S ′(Rn) is continuous, which com-

pletes the proof of Lemma 2.5.

For a sequence t = tQQ∈Q∗ ⊂ C, r ∈ (0,∞) and λ ∈ (0,∞), let

(t∗r,λ)Q :=

∑

R∈Q∗, ℓ(R)=ℓ(Q)

|tR|r[1 + ℓ(R)−1|xR − xQ|]λ

1r

, Q ∈ Q∗,

and t∗r,λ := (t∗r,λ)QQ∈Q∗ .We have the following estimates.

Lemma 2.8. Let p, q, s and φ be as in Definition 1.4, r ∈ (0,minp−, q−) andλ ∈ (n + Clog(s) + r log2(c1c1),∞),

where c1 and c1 are as in the condition (S1) and Clog(s) is as in (1.1) withg replaced by s. Then there exists a constant C ∈ [1,∞) such that, for all

t ∈ fs(·),φp(·),q(·)(R

n),

‖t‖fs(·),φp(·),q(·)

(Rn)≤ ‖t∗r,λ‖fs(·),φ

p(·),q(·)(Rn)

≤ C‖t‖fs(·),φp(·),q(·)

(Rn).

To prove Lemma 2.8, we need Lemma 2.9 below, which is just [20, Theorem3.2] (the vector-valued convolution inequality) and plays a key role throughoutthis article. In what follows, for any m ∈ (0,∞) and j ∈ Z, let

ηj,m(x) := 2jn(1 + 2j |x|)−m, x ∈ Rn.

Lemma 2.9. Let p, q ∈ C log(Rn) satisfy 1 < p− ≤ p+ < ∞ and 1 < q− ≤ q+ <∞. Let m ∈ (n,∞). Then there exists a positive constant C such that, for allsequences fjj∈N ⊂ L1

loc (Rn),

∥∥∥∥∥∥

∑

j∈N|ηj,m ∗ fj|q(·)

1q(·)

∥∥∥∥∥∥Lp(·)(Rn)

≤ C

∥∥∥∥∥∥

∑

j∈N|fj |q(·)

1q(·)

∥∥∥∥∥∥Lp(·)(Rn)

.

The following Lemma 2.10 is just [36, Lemma 19] (see also [20, Lemma 6.1]).

Lemma 2.10. Let s ∈ C logloc (R

n) and L ∈ [Clog(s),∞), where Clog(s) is as in(1.1) with g replaced by s. Then there exists a positive constant C such that, forall x, y ∈ Rn, m ∈ (0,∞) and v ∈ Z+,

2vs(x)ηv,m+L(x− y) ≤ C2vs(y)ηv,m(x− y).


Proof of Lemma 2.8. Notice that, for all Q ∈ Q∗, |tQ| ≤ (t∗r,λ)Q. This immedi-

ately implies that ‖t‖fs(·),φp(·),q(·)

(Rn)≤ ‖t∗r,λ‖fs(·),φ

p(·),q(·)(Rn)

, since fs(·),φp(·),q(·)(R

n) is a quasi-

Banach lattice (see Remark 2.2(i)).Conversely, let P be a given dyadic cube. For any Q ∈ Q∗, let vQ := tQ if

Q ⊂ 3P and vQ := 0 otherwise, and let uQ := tQ − vQ. Set v := vQQ∈Q∗ andu := uQQ∈Q∗. Then, for all Q ∈ Q∗, we have

(t∗r,λ)Q . (v∗r,λ)Q + (u∗r,λ)Q. (2.4)

From the proof of [20, Theorem 3.11], we deduce that, for all t ∈ fs(·),1p(·),q(·)(R

n),

‖t∗r,λ‖fs(·),1p(·),q(·)

(Rn). ‖t‖

fs(·),1p(·),q(·)

(Rn),

which implies that

IP :=1

φ(P )

∥∥∥∥∥∥

[∑

Q⊂P,Q∈Q∗

|Q|−[

s(·)n

+ 12](v∗r,λ)QχQ

q(·)] 1

q(·)

∥∥∥∥∥∥Lp(·)(P )

.1

φ(P )

∥∥v∗r,λ∥∥fs(·),1p(·),q(·)

(Rn).

1

φ(P )‖v‖

fs(·),1p(·),q(·)

(Rn)

∼ 1

φ(P )

∥∥∥∥∥∥

[∑

Q⊂3P,Q∈Q∗

|Q|−[

s(·)n

+ 12]|tQ|χQ

q(·)] 1

q(·)

∥∥∥∥∥∥Lp(·)(3P )

.

By this and the condition (S1) of φ, we conclude that

‖v∗r,λ‖fs(·),φp(·),q(·)

(Rn)≤ sup

P∈QIP . ‖t‖

fs(·),φp(·),q(·)

(Rn). (2.5)

Next, we deal with u. To this end, let, for i ∈ Z+ and k ∈ Zn,

A(i, k, P ) :=Q ∈ Q∗ : ℓ(Q) = 2−iℓ(P ), Q ⊂ P + kℓ(P )

.

Then, we have

JP :=1

φ(P )

∥∥∥∥∥∥

[∑

Q⊂P,Q∈Q∗

|Q|−[ s(·)

n+ 1

2](u∗r,λ)QχQ

q(·)] 1

q(·)

∥∥∥∥∥∥Lp(·)(P )

=1

φ(P )

∥∥∥∥∥∥∥

∞∑

i=0

∑

Q⊂P,Q∈Q∗

ℓ(Q)=2−iℓ(P )

(|Q|−[

s(·)n

+ 12]

×

∑

k∈Zn

|k|≥2

∑

R∈A(i,k,P )

|uR|r[1 + ℓ(Q)−1|xR − xQ|]λ

1r

χQ

q(·)

1q(·)

∥∥∥∥∥∥∥∥∥∥Lp(·)(P )

.

Notice that, when x ∈ Q, y ∈ R and ℓ(R) = ℓ(Q),

1 + [ℓ(Q)]−1|x− y| ∼ 1 + [ℓ(Q)]−1|xQ − xR|.


From this, we deduce that, for all x ∈ Q, µ ∈ (0,∞), j ∈ Z+ and k ∈ Zn,

ηjQ,µ ∗

∑

R∈A(i,k,P )

|uR|χR

r (x) =

∑

R∈A(i,k,P )

∫

R

2njQ|uR|r(1 + 2jQ|x− y|)µ dy

∼∑

R∈A(i,k,P )

|uR|r[1 + ℓ(Q)−1|xQ − xR|]µ

.

Since

λ > n+ Clog(s) + r log2(c1c1),

it follows that there exist m ∈ (n,∞) and L ∈ (Clog(s),∞) such that

λ > m+ L+ r log2(c1c1).

Observe that, when |k| ≥ 2, i ∈ Z+, Q ⊂ P with ℓ(Q) = 2−iℓ(P ) and R ∈A(i, k, P ),

1 + [ℓ(Q)]−1|xQ − xR| ∼ 2i|k|.Then, by Lemma 2.10, we conclude that

JP .1

φ(P )

∥∥∥∥∥∥∥

∞∑

i=0

∑

ℓ(Q)=2−iℓ(P )Q∈Q∗, Q⊂P

∑

k∈Zn, |k|≥2

(2i|k|)m+L−λ|Q|−[ s(·)n

+ 12]r

× ηjQ,m+L ∗

∑

R∈A(i,k,P )

|uR|χR

r

1r

χQ

q(·)

1q(·)

∥∥∥∥∥∥∥∥∥Lp(·)(P )

.1

φ(P )

∥∥∥∥∥∥

∞∑

i=0

∑

k∈Zn, |k|≥2

(2i|k|)m+L−λ

× ηi+jP ,m ∗

∑

R∈A(i,k,P )

|R|−[ s(·)n

+ 12]|uR|χR

r

q(·)r

1q(·)

∥∥∥∥∥∥∥∥Lp(·)(P )

,

which, combined with Remark 1.1(i) and the fact that, for all d ∈ [0, 1] andθjj ⊂ C,

(∑

j

|θj |)d

≤∑

j

|θj |d, (2.6)

implies that

JP .1

φ(P )

∥∥∥∥∥∥

∞∑

i=0

∑

k∈Zn, |k|≥2

(2i|k|)m+L−λ


× ηi+jP ,m ∗

∑

R∈A(i,k,P )

|R|−[s(·)n

+ 12]|uR|χR

r∥∥∥∥∥∥

1r

Lp(·)r (P )

.1

φ(P )

∞∑

i=0

∑

k∈Zn, |k|≥2

(2i|k|)m+L−λ

×

∥∥∥∥∥∥ηi+jP ,m ∗

∑

R∈A(i,k,P )

|R|−[s(·)n

+ 12]|uR|χR

r∥∥∥∥∥∥L

p(·)r (P )

1r

.

From this, m ∈ (n,∞), Lemma 2.9, Lemma 2.6(ii) and

λ > m+ L+ r log2(c1c1),

we deduce that

JP .1

φ(P )

∞∑

i=0

∑

k∈Zn

|k|≥2

(2i|k|)m+L−λ

∥∥∥∥∥∥

∑

R∈A(i,k,P )

|R|−[ s(·)n

+ 12]|uR|χR

∥∥∥∥∥∥

r

Lp(·)(Rn)

1r

.

∞∑

i=0

∑

k∈Zn

|k|≥2

(2i|k|)m+L−λ[φ(P + kℓ(P ))

φ(P )

]r

1r

‖t‖fs(·),φp(·),q(·)

(Rn)

.

∞∑

i=0

∑

k∈Zn

|k|≥2

2i(m+L−λ)|k|m+L−λ|k|r log(c1c1)

1r

‖t‖fs(·),φp(·),q(·)

(Rn)∼ ‖t‖

fs(·),φp(·),q(·)

(Rn).

This, together with the arbitrary of P ∈ Q, further implies that

‖u∗r,λ‖fs(·),φp(·),q(·)

(Rn). ‖t‖

fs(·),φp(·),q(·)

(Rn). (2.7)

Combining (2.4), (2.5) and (2.7), we conclude that

‖t∗r,λ‖fs(·),φp(·),q(·)

(Rn). ‖t‖

fs(·),φp(·),q(·)

(Rn),

which completes the proof of Lemma 2.8.

Now we give the proof of Theorem 2.3.

Proof of Theorem 2.3. We first show that Sϕ is bounded from Fs(·),φp(·),q(·)(R

n) to

fs(·),φp(·),q(·)(R

n). Let f ∈ Fs(·),φp(·),q(·)(R

n). From [20, Lemma A.6] (the r-trick lemma)

and its proof, we deduce that, for all L ∈ [Clog(s),∞), m ∈ (n + log2 c1,∞),

r ∈ (0,min1, p−, q−)and x ∈ Q := Qjk ∈ Q∗,

supz∈Q

|ϕj ∗ f(z)|r . 2jn∑

l∈Zn

∫

Qj(k+l)

(1 + 2j|x− y|)−(2m+L)|ϕj ∗ f(y)|r dy,


which implies that

‖Sϕf‖fs(·),φp(·),q(·)

(Rn)

. supP∈Q

1

φ(P )

∥∥∥∥∥∥

∞∑

j=(jP∨0)

∑

k∈Zn

2js(·)

×[2jn∑

l∈Zn

∫

Qj(k+l)

|ϕj ∗ f(y)|r(1 + 2j| · −y|)2m+L

dy

]1r

χQjk

q(·)

1q(·)

∥∥∥∥∥∥∥∥Lp(·)(P )

. supP∈Q

1

φ(P )

∥∥∥∥∥∥

∞∑

j=(jP∨0)

∑

k∈Zn

2js(·)∑

l∈Zn

(1 + |l|)−m

×[ηj,m+L ∗ (|ϕj ∗ fχQj(k+l)

|r)]χQjk

q(·)r

1q(·)

∥∥∥∥∥∥∥Lp(·)(P )

, (2.8)

where the last inequality comes from the fact that, when x ∈ Qjk and y ∈ Qj(k+l),

1 + 2j|x− y| ∼ 1 + |l|.

Observe that, for any given P ∈ Q, if Qjk ⊂ P , then Qj(k+l) ⊂ 3n|l|P for alll ∈ Zn. By this and (2.8), we see that

‖Sϕf‖fs(·),φp(·),q(·)

(Rn). sup

P∈Q

1

φ(P )

∥∥∥∥∥∥

∞∑

j=(jP∨0)

2jrs(·)

×[∑

l∈Zn

(1 + |l|)−mηj,m+L ∗ (|ϕj ∗ fχ3n|l|P |r)] q(·)

r

1q(·)

∥∥∥∥∥∥∥Lp(·)(P )

,

which, combined with the Minkowski inequality, Lemma 2.10 and Remark 1.1(i),further implies that

‖Sϕf‖fs(·),φp(·),q(·)

(Rn)

. supP∈Q

1

φ(P )

∑

l∈Zn

(1 + |l|)−m

×

∥∥∥∥∥∥∥

∞∑

j=(jP∨0)

[ηj,m+L ∗ (|2js(·)ϕj ∗ fχ3n|l|P |r)

] q(·)r

rq(·)

∥∥∥∥∥∥∥L

p(·)r (P )

1r

.


From this, Lemma 2.9, m ∈ (n+log2 c1,∞) and the condition (S1) of φ, it followsthat

‖Sϕf‖fs(·),φp(·),q(·)

(Rn)

. supP∈Q

1

φ(P )

∑

l∈Zn

(1 + |l|)−m

∥∥∥∥∥∥∥

∞∑

j=(jP∨0)

(2js(·)|ϕj ∗ f |

)q(·)

1q(·)

∥∥∥∥∥∥∥

r

Lp(·)(3n|l|P )

1r

. ‖f‖F

s(·),φp(·),q(·)

(Rn)

[∑

l∈Zn

(1 + |l|)−m(1 + |l|)log2 c1] 1

r

. ‖f‖F

s(·),φp(·),q(·)

(Rn),

which implies that Sϕ is bounded from Fs(·),φp(·),q(·)(R

n) to fs(·),φp(·),q(·)(R

n).

By repeating the argument used in the proof of [80, Theorem 2.1], with [80,Lemmas 2.7 and 2.8] therein replaced by Lemmas 2.5 and 2.8 here, we conclude

that Tψ is bounded from fs(·),φp(·),q(·)(R

n) to Fs(·),φp(·),q(·)(R

n), the details being omitted.

Finally, by the Calderon reproducing formula (see, for example, [80, Lemma 2.3]),

we know that Tψ Sϕ is the identity on Fs(·),φp(·),q(·)(R

n), which completes the proof

of Theorem 2.3.

3. Several equivalent characterizations of Fs(·),φp(·),q(·)(R

n)

In this section, we first establish molecular and atomic characterizations for

Fs(·),φp(·),q(·)(R

n) via Sobolev embeddings. Secondly, we characterize Fs(·),φp(·),q(·)(R

n) in

terms of the Peetre maximal function, which is further applied to show that

S(Rn) → Fs(·),φp(·),q(·)(R

n) → S ′(Rn)

and give out two equivalent quasi-norms of Fs(·),φp(·),q(·)(R

n), which may be useful in

applications.For notational simplicity, in what follows, for all Q ∈ Q∗, let χQ := |Q|−1/2χQ.

Proposition 3.1. Let φ be a set function as in Definition 1.4, s0, s1, p0, p1 bemeasurable functions satisfying that, for all x ∈ Rn, −∞ < s1(x) ≤ s0(x) < ∞,0 < p0(x) ≤ p1(x) <∞ and

s0(x)−n

p0(x)= s1(x)−

n

p1(x).

Assume that 0 < (p0)− ≤ (p1)− ≤ (p1)+ <∞ and s0,1p0

∈ C logloc (R

n). If q(x) = ∞for all x ∈ Rn or 0 < q− ≤ q(x) <∞ for all x ∈ Rn, then

fs0(·),φp0(·),q(·)(R

n) → fs1(·),φp1(·),q(·)(R

n).

Proof. Let t := tQQ∈Q∗ ∈ fs0(·),φp0(·),q(·)(R

n). We need to prove

‖t‖fs1(·),φ

p1(·),q(·)(Rn)

. ‖t‖fs0(·),φ

p0(·),q(·)(Rn)

.


To this end, let P ∈ Q be any given dyadic cube. For all Q ∈ Q∗, let uQ := tQwhen Q ⊂ P and uQ := 0 otherwise. Then, by the Sobolev embedding theorem

([72, Theorem 3.1]), namely, fs0(·),1p0(·),q(·)(R

n) → fs1(·),1p1(·),q(·)(R

n), we conclude that

∥∥∥∥∥∥∥

∞∑

j=(jP∨0)

∑

k∈Zn

[2js1(·)|tQjk

|χQjk

]q(·)

1q(·)

∥∥∥∥∥∥∥Lp1(·)(P )

=

∥∥∥∥∥∥

∞∑

j=0

∑

k∈Zn

[2js1(·)|uQjk

|χQjk

]q(·) 1

q(·)

∥∥∥∥∥∥Lp1(·)(Rn)

= ‖u‖fs1(·),1

p1(·),q(·)(Rn)

. ‖u‖fs0(·),1

p0(·),q(·)(Rn)

∼

∥∥∥∥∥∥

∞∑

j=0

∑

k∈Zn

[2js0(·)|uQjk

|χQjk

]q(·) 1

q(·)

∥∥∥∥∥∥Lp0(·)(Rn)

∼

∥∥∥∥∥∥∥

∞∑

j=(jP∨0)

∑

k∈Zn

[2js0(·)|tQjk

|χQjk

]q(·)

1q(·)

∥∥∥∥∥∥∥Lp0(·)(P )

,

which implies that

‖t‖fs1(·),φ

p1(·),q(·)(Rn)

. supP∈Q

1

φ(P )

∥∥∥∥∥∥∥

∞∑

j=(jP∨0)

∑

k∈Zn

[2js0(·)|tQjk

|χQjk

]q(·)

1q(·)

∥∥∥∥∥∥∥Lp0(·)(P )

∼‖t‖fs0(·),φ

p0(·),q(·)(Rn)

.

This finishes the proof of Proposition 3.1

Proposition 3.2. Let φ be a set function as in Definition 1.4, s0, s1, p0, p1 bemeasurable functions satisfying that, for all x ∈ Rn, −∞ < s1(x) ≤ s0(x) < ∞,0 < p0(x) ≤ p1(x) <∞ and

s0(x)−n

p0(x)= s1(x)−

n

p1(x).

Assume that 0 < (p0)− ≤ (p1)+ < ∞, s0,1p0

∈ C logloc (R

n) and infx∈Rn [s0(x) −s1(x)] > 0. Then, for all q ∈ (0,∞],

fs0(·),φp0(·),∞(Rn) → f

s1(·),φp1(·),q (R

n).

The proof of Proposition 3.2 is similar to that of Proposition 3.1, with [72,Theorem 3.1] replaced by [72, Theorem 3.2], the details being omitted.

Remark 3.3. (i) When φ(Q) = 1 for all cube Q, the conclusions of Propositions3.1 and 3.2 are just [72, Theorem 3.1] and [72, Theorem 3.2], respectively.

(ii) When p, q, s and φ are as in Remark 1.5(ii), Proposition 3.2 goes back to[80, Proposition 2.5].


Combining Theorem 2.3 and Proposition 3.1, we immediately obtain the fol-lowing Corollary 3.4, the details being omitted.

Corollary 3.4. Let i ∈ 0, 1, pi, q ∈ P(Rn) satisfy 1pi, 1

q∈ C log(Rn) and si

be measurable functions satisfying si ∈ C logloc (R

n) ∩ L∞(Rn), and φ a set functionsatisfying the conditions (S1) and (S2). Under the same assumptions as inProposition 3.1, the following conclusion

Fs0(·),φp0(·),q(·)(R

n) → Fs1(·),φp1(·),q(·)(R

n)

holds true.

Corollary 3.5. Let i ∈ 0, 1, pi, qi ∈ P(Rn) satisfy 1pi, 1

qi∈ C log(Rn) and si

be measurable functions satisfying si ∈ C logloc (R

n) ∩ L∞(Rn), and φ a set functionsatisfying the conditions (S1) and (S2). Assume that, for all x ∈ Rn,

s0(x)−n

p0(x)= s1(x)−

n

p1(x)

and infx∈Rn [s0(x)− s1(x)] > 0. Then

Fs0(·),φp0(·),q0(·)(R

n) → Fs1(·),φp1(·),q1(·)(R

n).

Proof. By Proposition 3.2 and (2.6), we see that

fs0(·),φp0(·),q0(·)(R

n) → fs0(·),φp0(·),∞(Rn) → f

s1(·),φp1(·),(q1)−(R

n) → fs1(·),φp1(·),q1(·)(R

n).

From this and Theorem 2.3, we deduce that Fs0(·),φp0(·),q0(·)(R

n) → Fs1(·),φp1(·),q1(·)(R

n),

which completes the proof of Corollary 3.5.

Next we establish molecular and atomic characterizations of Triebel-Lizorkin-type spaces with variable exponents.

Definition 3.6. Let K ∈ Z+, L ∈ Z and R ∈ N.

(i) A measurable function mQ on Rn is called a (K,L,R)-smooth moleculewith Q := Qjk ∈ Q, where j ∈ Z and k ∈ Zn, if it satisfies the followingconditions:

(M1) (vanishing moment) when j ∈ N,∫Rn x

γmQ(x) dx = 0 for all γ ∈ Zn+and |γ| < L;

(M2) (smoothness condition) for all multi-indices α ∈ Zn+, with |α| ≤ K,and all x ∈ Rn

|DαmQ(x)| ≤ 2(|α|+n/2)j(1 + 2j|x− xQ|)−R.(ii) A measurable function aQ on Rn is called a (K,L)-smooth atom supported

near Q := Qjk ∈ Q, where j ∈ Z and k ∈ Zn, if it satisfies the followingconditions:(A1) supp aQ ⊂ 3Q;(A2) (vanishing moment) when j ∈ N,

∫Rn x

γaQ(x) dx = 0 for all γ ∈ Zn+with |γ| < L;

(A3) (smoothness condition) for all multi-indices α ∈ Zn+ with |α| ≤ K,

|DαaQ(x)| ≤ 2(|α|+n/2)j for all x ∈ Rn.


Remark 3.7. (i) If L < 0, then the vanishing moment conditions (M1) and (A2)are avoid.

(ii) Let aQ be a (K,L)-smooth atom with Q := Qjk ∈ Q∗ with j ∈ Z+ andk ∈ Zn. Then, by combining the conditions (A1) and (A3), we conclude that, forall R ∈ (0,∞), α ∈ Zn+ with |α| ≤ K and x ∈ Rn,

|DαaQ(x)| ≤ C2(|α|+n/2)j1

(1 + 2j|x− xQ|)R,

where C is a positive constant independent of x, α, Q and aQ, but dependingon R. Thus, each (K,L)-smooth atom is a (K,L,R)-smooth molecule up to aharmless positive constant.

Theorem 3.8. Let p, q, s and φ be as in Definition 1.4.

(i) Let K ∈ (s+ +max0, log2 c1,∞) and

L ∈(

n

min1, p−, q−− n− s−,∞

). (3.1)

Suppose that mQQ∈Q∗ is a family of (K,L,R)-smooth molecules with

R large enough and that t := tQQ∈Q∗ ∈ fs(·),φp(·),q(·)(R

n). Then f :=∑Q∈Q∗ tQmQ converges in S ′(Rn) and

‖f‖F

s(·),φp(·),q(·)

(Rn)≤ C‖t‖

fs(·),φp(·),q(·)

(Rn)

with C being a positive constant independent of t.

(ii) Conversely, if f ∈ Fs(·),φp(·),q(·)(R

n), then, for any given K, L ∈ Z+, there exist

a sequence t := tQQ∈Q∗ ⊂ C and a sequence aQQ∈Q∗ of (K,L)-smoothatoms such that f =

∑Q∈Q∗ tQaQ in S ′(Rn) and

‖t‖fs(·),φp(·),q(·)

(Rn)≤ C‖f‖

Fs(·),φp(·),q(·)

(Rn)

with C being a positive constant independent of f .

Remark 3.9. (i) When φ(P ) := 1 for all cubes P ⊂ Rn, conclusions of Theorem3.8 coincide with those of [35, Corollary 5.6]; when p, q, s and φ are as in Remark1.5(ii), Theorem 3.8 goes back to [23, Theorem 3.12] (see also [80, Theorem 3.3]).

(ii) In the case that φ(P ) := 1 for all cubes P ⊂ Rn and s ≥ 0, the vanishingmoment and the smoothness conditions of Theorem 3.8 can be further refined.Indeed, it was proved in [20, Theorem 3.11] that the vanishing moment and thesmoothness conditions of atoms can be localized on dyadic cubes associated withatoms in Theorem 3.8.

Proof of Theorem 3.8. The proof of (ii) is similar to that of [80, Theorme 3.3](see also [27, Theorem 4.1]). Indeed, by repeating the argument that used in theproof of [80, Theorem 3.3], with [80, Lemma 2.8] therein replaced by Lemma 2.8,we can prove (ii), the details being omitted.

Next we prove (i) by two steps.


Step 1) We show that f =∑

Q∈Q∗ tQmQ converges in S ′(Rn). To this end, itsuffices to show that

limN→∞,Λ→∞

N∑

j=0

∑

k∈Zn, |k|≤Λ

tQjkmQjk

(3.2)

exists in S ′(Rn). For all h ∈ S(Rn) and j ∈ Z+, by the vanishing momentcondition (M1), we see that∫

Rn

∑

k∈Zn, |k|≤Λ

tQjkmQjk

(y)h(y) dy

=

∫

Rn

∑

k∈Zn, |k|≤Λ

tQjkmQjk

(y)

h(y)−

∑

γ∈Zn, |γ|<L(y − xQjk

)γDγh(xQjk

)

γ!

dy

and, by Taylor’s remainder theorem, we find that, for all y ∈ Rn,∣∣∣∣∣∣h(y)−

∑

γ∈Zn, |γ|≤L(y − xQjk

)γDγh(xQjk

)

γ!

∣∣∣∣∣∣

. |y − xQjk|L∑

|γ|=L

|Dγh(ξ(y))|γ!

. (1 + |y − xQjk|)L(1 + |y|)−δ sup

ξ∈Rn

∑

|γ|=L(1 + |ξ|)δ |D

γh(ξ)|γ!

,

where δ ∈ (0,∞) and ξ(y) := y + θ(xQjk− y) with some θ ∈ (0, 1) depending on

y and xQjk, which, together with the fact that, for all y ∈ Rn,

|mQjk(y)| ≤ 2jn/2(1 + 2j |y − xQjk

|)−R,further implies that

∣∣∣∣∣∣

∫

Rn

∑

k∈Zn, |k|≤Λ

tQjkmQjk

(y)h(y) dy

∣∣∣∣∣∣

.

∫

Rn

∑

k∈Zn, |k|≤Λ

|tQjk|2−j(L−n/2) (1 + |y|)−δ

(1 + 2j|y − xQjk|)R−L dy

.

∞∑

v=0

∫

Dv

∑

k∈Zn

|tQjk|2−j(L−n/2) (1 + |y|)−δ

(1 + 2j|y − xQjk|)R−L dy, (3.3)

where D0 := x ∈ Rn : |x| ≤ 1 and, for all v ∈ N,

Dv := x ∈ Rn : 2v−1 < |x| ≤ 2v.For all v ∈ Z+, y ∈ Dv and j ∈ Z+, let W

y,j0 := k ∈ Zn : 2j|y − xQjk

| ≤ 1 and,for i ∈ N,

W y,ji := k ∈ Zn : 2i−1 < 2j|y − xQjk

| ≤ 2i.


Then we have

H(v, j, y) :=∑

k∈Zn

|tQjk||Qjk|−

12 (1 + 2j|y − 2−jk|)−(R−L)

∼∞∑

i=0

∑

k∈W y,ji

2−i(R−L)|tQjk||Qjk|−

12

∼∞∑

i=0

2−i(R−L)∫

∪k∈W

y,ji

Qjk

2jn

[∑

k∈Zn

|tQjk|χQjk

(z)

]dz.

Observe that, if z ∈ ∪k∈W y,jiQjk, then z ∈ Qjk0

for some k0 ∈ W y,ji and, for

y ∈ Dv, 1 + 2j |y − z| ∼ 1 + 2i; moreover,

|z| ≤ |z − xQjk0|+ |xQjk0 − y|+ |y| . 2−j + 2−j+i + 2v . 2i+v,

which implies that ⋃

k∈W y,ji

Qjk ⊂ Q(0, 2i+v+c0)

with some positive constant c0 ∈ N. From this, we deduce that, for all a ∈ (n,∞),v, j ∈ Z+ and y ∈ Rn,

H(v, j, y)∼∞∑

i=0

2−i(R−L−a)∫

∪k∈W

y,ji

Qjk

2jn

(1 + 2j |y − z|)a

×[∑

k∈Zn

|tQjk|χQjk

(z)χQ(0,2i+v+c0 )(z)

]dz

.

∞∑

i=0

2−i(R−L−a)ηj,a ∗(∑

k∈Zn

|tQjk|χQjk

χQ(0,2i+v+c0)

)(y),

which, combined with (3.3), implies that∣∣∣∣∣∣

∫

Rn

∑

k∈Zn, |k|≤Λ

tQjkmQjk

(y)h(y) dy

∣∣∣∣∣∣

. 2−jL∞∑

v=0

∞∑

i=0

(1 + 2v)−δ02−i(R−L−a)

×∫

Dv

ηj,a ∗(∑

k∈Zn

|tQjk|χQjk

χQ(0,2i+v+c0 )

)(y)(1 + |y|)−δ+δ0 dy, (3.4)

where δ0 ∈ (0,∞) is determined later.By (3.1), we find that there exists r ∈ (0,min1, p−, q−) such that

s− +n

p−(r − 1) > −L.


Let, for all x ∈ Rn, p(x) := p(x)/r, (p(x))∗ := p(x)p(x)−1

and s be a measurable

function on Rn such that, for all x ∈ Rn,

s(x)− n

p(x)= s(x)− n

p(x).

Then

s− = infx∈Rn

s(x) +

n(r − 1)

p(x)

≥ inf

x∈Rn[s(x)] + inf

x∈Rn

[n(r − 1)

p(x)

]

= s− +n

p−(r − 1) > −L.

Choosing δ ∈ (0,∞) and δ0 ∈ (max0, log2 c1,∞) such that δ ∈ (n(1− r)/p+ +δ0,∞), by the Holder inequality in Remark 1.1(iii), (3.4), Lemma 2.9, Remark2.2(ii) and Proposition 3.1, we conclude that

∣∣∣∣∣∣

∫

Rn

∑

k∈Zn, |k|≤Λ

tQjkmQjk

(y)h(y) dy

∣∣∣∣∣∣

. 2−j(L+s−)∞∑

v=0

2−vδ0∞∑

i=0

2−i(R−L−a)∥∥(1 + | · |)−δ+δ0

∥∥L(p(·))∗(Rn)

×∥∥∥∥∥ηj,a ∗

[∑

k∈Zn

2js(·)|tQjk|χQjk

χQ(0,2i+v+c0)

]∥∥∥∥∥Lp(·)(Rn)

. 2−j(L+s−)

∞∑

v=0

2−vδ0∞∑

i=0

2−i(R−L−a)

∥∥∥∥∥∑

k∈Zn

2js(·)|tQjk|χQjk

∥∥∥∥∥Lp(·)(Q(0,2i+v+c0))

. 2−j(L+s−)∞∑

v=0

2−vδ0∞∑

i=0

2−i(R−L−a)φ(Q(0, 2i+v+c0))‖t‖fs(·),φp(·),q(·)

(Rn)

. 2−j(L+s−)∞∑

v=0

2−v(δ0−log2 c1)∞∑

i=0

2−i(R−L−a−log2 c1)‖t‖fs(·),φp(·),q(·)

(Rn)

. 2−j(L+s−)‖t‖fs(·),φp(·),q(·)

(Rn),

where R ∈ (0,∞) is chosen large enough, which, together with L > −s−, impliesthat (3.2) exists in S ′(Rn) and |〈f, h〉| . ‖t‖

fs(·),φp(·),q(·)

(Rn).

Step 2 ) We prove that ‖f‖F

s(·),φp(·),q(·)

(Rn). ‖t‖

fs(·),φp(·),q(·)

(Rn). Let P ∈ Q be a given

dyadic cube and r ∈ (0,min1, p−, q−) such that L > n/r − n − s−. Then, byRemark 1.1(i), we find that

1

φ(P )

∥∥∥∥∥∥∥

∞∑

j=(jP∨0)

[2js(·)|ϕj ∗ f |

]q(·)

1q(·)

∥∥∥∥∥∥∥Lp(·)(P )


×∞∑

i=0

2−Rri+vn

∫

∪k∈Ω

x,vi

Qvk

∑

k∈Ωx,vi

|tQvk|χQvk

(y)

r

dy

. (3.6)

Observe that, if y ∈ ∪k∈Ωx,viQvk, then there exists a k0 ∈ Ωx,vi such that y ∈ Qvk0

and 1 + 2v|x− y| ∼ 1 + 2i; moreover, since v ≤ jP , it follows that

|y − cP | ≤ |y − xQvk0

|+ |x− xQvk0

|+ |x− cP |. 2−v + 2i−v + 2−jP . 2i−v,

which implies that ∪k∈Ωx,viQvk ⊂ Q(cP , 2

i−v+c0) for some constant c0 ∈ N. From

this, (3.6) and Lemma 2.10, we deduce that, for all a ∈ (n/r,∞), v, j ∈ Z+ andx ∈ P ,

J(v, j, x, P ) . 2js(x)r2(v−j)Kr∞∑

i=0

2(a+ε/r−R)ri∫

∪k∈Ω

x,vi

Qvk

2vn

(1 + 2v|x− y|)ar+ε

×

∑

k∈Ωx,vi

|tQvk|χQvk

χQ(cP ,2i−v+c0)(y)

r

dy

. 2(v−j)(K−s+)r∞∑

i=0

2(a+ε/r−R)ri

×ηv,ar ∗

∑

k∈Ωx,vi

2vs(·)|tQvk|χQvk

χQ(cP ,2i−v+c0)

r (x),

which implies that the claim holds true.By this claim, (3.5) and Remark 1.1(i), we conclude that

I1.1

φ(P )

∞∑

j=jP

jP−1∑

v=0

2(v−j)(K−s+)r∞∑

i=0

2(a+ε/r−M)ri

×

∥∥∥∥∥∥ηv,ar ∗

∑

k∈Ω·,vi

|tQvk|2vs(·)χQvk

χB(cP ,2i−v+c0)

r∥∥∥∥∥∥L

p(·)r (P )

1r

,

which, together with Lemma 2.9, jP ∈ N and Remark 2.2(ii), further implies that

I1.1

φ(P )

∞∑

j=jP

jP∑

v=0

2(v−j)(K−s+)r∞∑

i=0

2(a+ε/r−R)ri

×∥∥∥∥∥∑

k∈Zn

|tQvk|2vs(·)χQvk

∥∥∥∥∥

r

Lp(·)(Q(cP ,2i−v+c0))

1r

. ‖t‖fs(·),φp(·),q(·)

(Rn)

∞∑

j=jP

jP∑

v=0

2(v−j)(K−s+)r


×∞∑

i=0

2(a+ε/r−R)ri[φ(Q(cP , 2

i−v+c0))]r

[φ(P )]r

1r

.

From this, K ∈ (s+ +max0, log2 c1,∞) and the fact that, when v ≤ jP ,

φ(Q(cP , 2i−v+c0)). (c1)

iφ(Q(cP , 2−v)) . (c1)

i+jP−vφ(P )

∼ 2(i+jP−v) log2 c1φ(P ),

we deduce that

I1. ‖t‖fs(·),φp(·),q(·)

(Rn)

2jP log2 c1

∞∑

j=jP

2−j(K−s+)r

jP∑

v=0

2v(K−s+−log2 c1)r

×∞∑

i=0

2−i(R−a−ε/r−log2 c1)r

1r

. ‖t‖fs(·),φp(·),q(·)

(Rn), (3.7)

where R ∈ (0,∞) is chosen such that R > a+ ε/r + log2 c1.We now estimate I2. By applying [20, Lemmas A.2 and A.5] and an argument

similar to that used in the proof of [20, Lemma 6.3], we see that, for all j ∈ Z+,Q := Qvk ∈ Q∗ and x ∈ Rn,

|ϕj ∗mQ(x)| . 2−β(j,v)|Q|−1/2(ηj,R ∗ ηv,R ∗ χQ)(x),where

β(j, v) := Kmaxj − v, 0+ Lmaxv − j, 0.Let

M ∈ (n/r + log2(c1c1),∞)

and ε ∈ [Clog(s),∞) be such that R = 2M + ε/r. Thus, we have

I2.1

φ(P )

∥∥∥∥∥∥

∞∑

j=(jP∨0)

∞∑

v=(jP∨0)

∑

ℓ(Q)=2−v

2js(·)r2−β(j,v)r

× |tQ|r|Q|−r2 (ηj,2M+ε/r ∗ ηv,2M+ε/r ∗ χQ)r

] q(·)r

rq(·)

∥∥∥∥∥∥∥

1r

Lp(·)(P )

. (3.8)

By [20, Lemma A.4], we find that, for all v ∈ Z+ and ℓ(Q) = 2−v,

2js(·)r−β(j,v)r(ηj,2M+ε/r ∗ ηv,2M+ε/r ∗ χQ)r

. 2js(·)r−β(j,v)r2nmaxv−j,0(1−r)ηj,2Mr+ε ∗ ηv,2Mr+ε ∗ χQ∼ 2vs(·)r2−(K−s+)rmaxj−v,0

×2−(L−nr+n+s−)rmaxv−j,0ηj,2Mr+ε ∗ ηv,2Mr+ε ∗ χQ,

which, combined with (3.8) and Lemma 2.10, implies that

I2.1

φ(P )

∥∥∥∥∥∥

∞∑

j=(jP∨0)

ηj,2Mr ∗

∞∑

v=(jP∨0)

∑

ℓ(Q)=2−v

|tQ|r|Q|−r2


× 2vs(·)r−ε(j,v)rηv,2Mr+ε ∗ χQ]) q(·)

r

rq(·)

∥∥∥∥∥∥∥

1r

Lp(·)r (P )

.1

φ(P )

∥∥∥∥∥∥

∞∑

j=(jP∨0)

∑

l∈Zn

∫

P+lℓ(P )

2jn

(1 + 2j| · −y|)2Mr

∞∑

v=(jP∨0)2vs(·)−ε(j,v)

×∑

ℓ(Q)=2−v

[|tQ||Q| 12

]rηv,2Mr+ε ∗ χQ

(y) dy

q(·)r

rq(·)

∥∥∥∥∥∥∥∥

1r

Lp(·)r (P )

, (3.9)

where

ε(j, v) := (K − s+)maxj − v, 0+ (L− n/r + n+ s−)maxv − j, 0.From this, the fact that, when j ≥ jP , l ∈ Zn, x ∈ P and y ∈ P + lℓ(P ),

1 + 2j|x− y| ≥ 1 + 2jP |x− y| ∼ 1 + |l|,the Minkowski inequality, Lemma 2.9 and Remark 1.1(i), we further deduce that

I2 .1

φ(P )

∥∥∥∥∥∥

∞∑

j=(jP∨0)

∑

l∈Zn

(1 + |l|)−Mrηj,Mr ∗

∞∑

v=(jP∨0)

∑

ℓ(Q)=2−v

|tQ|r

× |Q|− r22vs(·)r−ε(j,v)rηv,2Mr+ε ∗ χQ

]χP+lℓ(P )

)] q(·)r

rq(·)

∥∥∥∥∥∥∥

1r

Lp(·)r (P )

.1

φ(P )

∥∥∥∥∥∥

∑

l∈Zn

(1 + |l|)−Mr

∞∑

j=(jP∨0)

ηj,Mr ∗

∞∑

v=(jP∨0)

∑

ℓ(Q)=2−v

|tQ|r

× |Q|− r22vs(·)r−ε(j,v)rηv,2Mr+ε ∗ χQ

]χP+lℓ(P )

)] q(·)r

rq(·)

∥∥∥∥∥∥∥

1r

Lp(·)r (P )

.1

φ(P )

∑

l∈Zn

(1 + |l|)−Mr

∥∥∥∥∥∥

∞∑

j=(jP∨0)

∞∑

v=(jP∨0)

∑

ℓ(Q)=2−v

|tQ|r|Q|−r2

× 2vs(·)r−ε(j,v)rηv,2Mr+ε ∗ χQ) q(·)

r

rq(·)

∥∥∥∥∥∥∥L

p(·)r (P+lℓ(P ))

1r

. (3.10)

By the Holder inequality,

K ∈ (s+ +max0, log2 c1,∞),


L ∈ (n/r − n− s−,∞) and the fact that 0 < q− ≤ q+ <∞, we see that

∞∑

j=(jP∨0)

∞∑

v=(jP∨0)

∑

ℓ(Q)=2−v

|tQ|r|Q|−r22vs(·)r−ε(j,v)rηv,2Mr+ε ∗ χQ

q(·)r

rq(·)

.

∞∑

j=(jP∨0)

∞∑

v=(jP∨0)2−ε(j,v)r

∑

ℓ(Q)=2−v

|tQ|r|Q|−r22vs(·)rηv,2Mr+ε ∗ χQ

q(·)r

rq(·)

.

∞∑

v=(jP∨0)

∑

ℓ(Q)=2−v

|tQ|r|Q|−r22vs(·)rηv,2Mr+ε ∗ χQ

q(·)r

rq(·)

,

which, together with (3.10) and some arguments similar to those used in theproofs of (3.9) and (3.10), implies that

I2.1

φ(P )

∑

l∈Zn

(1 + |l|)−Mr

∥∥∥∥∥∥

∞∑

v=(jP∨0)

∑

ℓ(Q)=2−v

|tQ|r|Q|−r2

×2vs(·)rηv,2Mr+ε ∗ χQ] q(·)

r

rq(·)

∥∥∥∥∥∥∥L

p(·)r (P+lℓ(P ))

1r

.1

φ(P )

∑

l∈Zn

(1 + |l|)−Mr

∥∥∥∥∥∥

∞∑

v=(jP∨0)

[∑

k∈Zn

(1 + |k|)−Mr

×ηv,Mr ∗

∑

ℓ(Q)=2−v

|tQ|2vs(·)χQ

r

×χP+(l+k)ℓ(P )

)] q(·)r

rq(·)

∥∥∥∥∥∥∥L

p(·)r (P+lℓ(P ))

1r

.

From this, the Minkowski inequality, Remark 1.1(i), Lemmas 2.9 and 2.6(ii), wededuce that

I2 .1

φ(P )

[∑

k,l∈Zn

(1 + |l|)−Mr(1 + |k|)−Mr

×

∥∥∥∥∥∥∥∥

∞∑

v=(jP∨0)

∑

ℓ(Q)=2−v

2vs(·)|tQ|χQ

q(·)

1q(·)

∥∥∥∥∥∥∥∥

r

Lp(·)(P+(l+k)ℓ(P ))

1r


. ‖t‖fs(·),φp(·),q(·)

(Rn)

∑

k,l∈Zn

(1 + |l|)−Mr(1 + |k|)−Mr [φ(P + (l + k)ℓ(P ))]r

[φ(P )]r

1r

. ‖t‖fs(·),φp(·),q(·)

(Rn)

∑

k,l∈Zn

(1 + |l|)−Mr+r log2(c1c1)(1 + |k|)−Mr+r log2(c1c1)

12

∼‖t‖fs(·),φp(·),q(·)

(Rn), (3.11)

where M is chosen large enough.Finally, combining (3.7) and (3.11), we conclude that

‖f‖F

s(·),φp(·),q(·)

(Rn). sup

P∈Q(I1 + I2) . ‖t‖

fs(·),φp(·),q(·)

(Rn),

which completes the proof of Theorem 3.8.

Next we establish the Peetre maximal function characterization of Fs(·),φp(·),q(·)(R

n).

Let (ϕ,Φ) be a pair of admissible functions. Recall that the Peetre maximalfunction of f ∈ S ′(Rn) is defined by setting, for all j ∈ Z+, a ∈ (0,∞) andx ∈ Rn,

(ϕ∗jf)a(x) := sup

y∈Rn

|ϕj ∗ f(x+ y)|(1 + 2j|y|)a ,

where ϕ0 is replaced by Φ. The following Lemma 3.10 comes from [71, (2.48) and(2.66)].

Lemma 3.10. Let (ϕ,Φ) be a pair of admissible functions, f ∈ S ′(Rn) andN ∈ N. Then, for all t ∈ [1, 2], a ∈ (0, N ], ℓ ∈ Z+ and x ∈ Rn,

[(ϕ∗2−ℓtf)a(x)]

r ≤ C

∞∑

v=0

2−vNr2(v+ℓ)n∫

Rn

|(ϕv+ℓ)t ∗ f(y)|r(1 + 2ℓ|x− y|)ar dy,

where r is an arbitrary fixed positive number, ϕ0 is replaced by Φ and C is apositive constant independent of ϕ, Φ, f , x, ℓ and t.

Theorem 3.11. Let p, q, s and φ be as in Definition 1.4. Let

a ∈(

n

minp−, q−+ log2 c1 + Clog(s),∞

). (3.12)

Then f ∈ Fs(·),φp(·),q(·)(R

n) if and only if f ∈ S ′(Rn) and ‖f‖∗F

s(·),φp(·),q(·)

(Rn)<∞, where

‖f‖∗F

s(·),φp(·),q(·)

(Rn):= sup

P∈Q

1

φ(P )

∥∥∥∥∥∥∥

∞∑

j=(jP∨0)

[2js(·)(ϕ∗

jf)a]q(·)

1q(·)

∥∥∥∥∥∥∥Lp(·)(P )

.

Proof. Observe that, by definitions, we have ‖f‖F

s(·),φp(·),q(·)

(Rn)≤ ‖f‖∗

Fs(·),φp(·),q(·)

(Rn). Next

we show that ‖f‖∗F

s(·),φp(·),q(·)

(Rn). ‖f‖

Fs(·),φp(·),q(·)

(Rn)for all f ∈ F

s(·),φp(·),q(·)(R

n).


By (3.12), we find that there exist r ∈ (0,minp−, q−) and ε ∈ (log2 c1,∞)such that a > n/r + ε+ Clog(s). For any given dyadic cube P ⊂ Rn, by Lemma3.10, we see that

JP :=1

φ(P )

∥∥∥∥∥∥∥

∞∑

j=(jP∨0)

[2js(·)(ϕ∗

jf)a]q(·)

1q(·)

∥∥∥∥∥∥∥Lp(·)(P )

.1

φ(P )

∥∥∥∥∥∥

∞∑

j=(jP∨0)

[2js(·)

( ∞∑

v=0

2−vNr2(v+j)n

×∫

Rn

|ϕv+j ∗ f(y)|r(1 + 2j| · −y|)ar dy

)1r

]q(·)

1q(·)

∥∥∥∥∥∥∥Lp(·)(P )

, (3.13)

where N ∈ N ∩ [a,∞) is determined later. Notice that j ≥ (jP ∨ 0) and, for allx ∈ P and y ∈ (2k+1

√nP )\(2k√nP ) =: Dk,P with k ∈ N,

1 + 2j|x− y| & 2j2−jP 2k.

Then it follows that, for all x ∈ P ,

∫

Rn

|ϕv+j ∗ f(y)|r(1 + 2j |x− y|)ar dy

=

∫

2√nP

+∞∑

k=1

∫

Dk,P

|ϕv+j ∗ f(y)|r

(1 + 2j |x− y|)ar dy

. 2−jnηj,ar ∗(|ϕv+j ∗ f |rχ2

√nP

)(x)

+2−j(εr+n)2jP εr∞∑

k=1

2−kεrηj,(a−ε)r ∗(|ϕv+j ∗ f |rχDk,P

)(x)

=: IP,1 + IP,2,

which implies that

JP .1

φ(P )

∥∥∥∥∥∥∥

∞∑

j=(jP∨0)

[2js(·)r

∑

v=0

2−vNr2(v+j)nIP,1

] q(·)r

1q(·)

∥∥∥∥∥∥∥Lp(·)(P )

+1

φ(P )

∥∥∥∥∥∥∥

∞∑

j=(jP∨0)

[2js(·)r

∑

v=0

2−vNr2(v+j)nIP,2

] q(·)r

1q(·)

∥∥∥∥∥∥∥Lp(·)(P )

=: JP,1 + JP,2. (3.14)


For JP,1, by Lemmas 2.10 and 2.9, the Minkowski inequality and Remark 1.1(i),we find that

JP,1.1

φ(P )

∥∥∥∥∥∥∥

∞∑

j=(jP∨0)

[ ∞∑

v=0

2(n−Nr)v2js(·)r|ϕv+j ∗ f |r] q(·)

r

rq(·)

∥∥∥∥∥∥∥

1r

Lp(·)r (2

√nP )

.

∞∑

v=0

2−v(Nr−n+s−r)1

[φ(P )]r

×

∥∥∥∥∥∥∥

∞∑

j=(jp∨0)

[2(v+j)s(·)|ϕv+j ∗ f |r

]q(·)

1q(·)

∥∥∥∥∥∥∥

r

Lp(·)(2√nP )

1r

.

∞∑

v=0

2−v(Nr−n+s−r)

1r

‖f‖F

s(·),φp(·),q(·)

(Rn)∼ ‖f‖

Fs(·),φp(·),q(·)

(Rn), (3.15)

where we used the condition (S1) of φ in the third inequality and N ∈ N is chosenlarge enough such that N ∈ [a,∞) ∩ (n

r− s−,∞).

For JP,2, by an argument similar to the above, we find that

JP,2.

∞∑

v=0

2−v(Nr−n+rs−)

∞∑

k=1

2−kε1

[φ(P )]r

×

∥∥∥∥∥∥∥

∞∑

j=(jp∨0)

[2(v+j)s(·)|ϕv+j ∗ f |r

]q(·)

1q(·)

∥∥∥∥∥∥∥

r

Lp(·)(Dk,P )

1r

.

∞∑

v=0

2−v(Nr−n+rs−)

∞∑

k=1

2−kεr[φ(2k+1+nP )]r

[φ(P )]r

1r

‖f‖F

s(·),φp(·),q(·)

(Rn)

.

∞∑

k=1

2−k(ε−log2 c1)

1r

‖f‖F

s(·),φp(·),q(·)

(Rn)∼ ‖f‖

Fs(·),φp(·),q(·)

(Rn). (3.16)

Combining the estimates (3.13), (3.14), (3.15) and (3.16), we conclude that

‖f‖∗F

s(·),φp(·),q(·)

(Rn)≤ sup

P∈QJP . sup

P∈Q(JP,1 + JP,2) . ‖f‖

Fs(·),φp(·),q(·)

(Rn),

which completes the proof of Theorem 3.11.

As applications of Theorem 3.11, we obtain two equivalent quasi-norms of the

space Fs(·),φp(·),q(·)(R

n). To this end, for all f ∈ S ′(Rn), let

∥∥∥f∣∣∣F s(·),φ

p(·),q(·)(Rn)∥∥∥1:= sup

P∈Q

1

φ(P )

∥∥∥∥∥∥

∞∑

j=0

[2js(·)|ϕj ∗ f |

]q(·) 1

q(·)

∥∥∥∥∥∥Lp(·)(P )


and ∥∥∥f∣∣∣F s(·),φ

p(·),q(·)(Rn)∥∥∥2:= sup

Q∈Qsupx∈Q

|Q|− s(x)n [φ(Q)]−1‖χQ‖Lp(·)(Rn)|ϕjQ ∗ f(x)|.

Theorem 3.12. Let p, q, s, φ be as in Definition 1.4.

(i) If c1 ∈ (0, 2n/p+), then f ∈ Fs(·),φp(·),q(·)(R

n) if and only if f ∈ S ′(Rn) and

‖f |F s(·),φp(·),q(·)(R

n)‖1 < ∞; moreover, there exists a positive constant C, in-

dependent of f , such that

C−1‖f‖F

s(·),φp(·),q(·)

(Rn)≤∥∥∥f∣∣∣F s(·),φ

p(·),q(·)(Rn)∥∥∥1≤ C‖f‖

Fs(·),φp(·),q(·)

(Rn).

(ii) If c1 ∈ (0, 2−n/p−), then f ∈ Fs(·),φp(·),q(·)(R

n) if and only if f ∈ S ′(Rn) and

‖f |F s(·),φp(·),q(·)(R

n)‖2 < ∞; moreover, there exists a positive constant C, in-

dependent of f , such that

C−1‖f‖F

s(·),φp(·),q(·)

(Rn)≤∥∥∥f∣∣∣F s(·),φ

p(·),q(·)(Rn)∥∥∥2≤ C‖f‖

Fs(·),φp(·),q(·)

(Rn).

Proof. We first show (i). To this end, it suffices to show that

‖f |F s(·),φp(·),q(·)(R

n)‖1 . ‖f‖F

s(·),φp(·),q(·)

(Rn),

since the inverse inequality obviously holds true by definitions.Let P ∈ Q be a given dyadic cube. By Remark 1.1(i), we see that

IP :=1

φ(P )

∥∥∥∥∥∥

∞∑

j=0

[2js(·)|ϕj ∗ f |

]q(·) 1

q(·)

∥∥∥∥∥∥Lp(·)(P )

.1

φ(P )

∥∥∥∥∥∥∥

(jP∨0)−1∑

j=0

[2js(·)|ϕj ∗ f |

]q(·)

1q(·)

∥∥∥∥∥∥∥Lp(·)(P )

+1

φ(P )

∥∥∥∥∥∥∥

∞∑

j=(jP∨0)

[2js(·)|ϕj ∗ f |

]q(·)

1q(·)

∥∥∥∥∥∥∥Lp(·)(P )

=: IP,1 + IP,2,

where∑(jP∨0−1)

j=0 · · · = 0 if jP ≤ 0.

Obviously, IP,2 ≤ ‖f‖F

s(·),φp(·),q(·)

(Rn).

For IP,1, we only need to estimate it in the case that jP > 0. For any j ∈ N

with j ≤ jP − 1, there exists a unique dyadic cube Pj such that P ⊂ Pj and

ℓ(Pj) = 2−j. Since s ∈ C logloc (R

n) ∩ L∞(Rn), it follows that, for all a ∈ (0,∞),x ∈ P and y ∈ Pj,

2js(x)|ϕj ∗ f(x)|. 2js(x)(1 + 2j|x− y|)a(ϕ∗jf)a(y)

. 2j[s(x)−s(y)]2js(y)(ϕ∗jf)a(y)

. 2jClog(s)

log(e+1/|x−y|)2js(y)(ϕ∗jf)a(y) . 2js(y)(ϕ∗

jf)a(y),


which implies that, for all x ∈ P ,

2js(x)|ϕj ∗ f(x)| . infy∈Pj

2js(y)(ϕ∗jf)a(y). (3.17)

Thus, choosing r ∈ (0,min1, p−, q−) and a as in Theorem 3.11, by Theorem3.11 and Remark 1.1(i), we conclude that

IP,1.1

φ(P )

∥∥∥∥∥∥

jP−1∑

j=0

[infy∈Pj

2js(y)(ϕ∗jf)a(y)

]q(·) 1q(·)

∥∥∥∥∥∥Lp(·)(P )

.1

φ(P )

∥∥∥∥∥

jP−1∑

j=0

∥∥2js(·)(ϕ∗jf)a

∥∥rLp(·)(Pj)

‖χPj‖−rLp(·)(Rn)

∥∥∥∥∥

1r

Lp(·)r (P )

. ‖f‖F

s(·),φp(·),q(·)

(Rn)

jP−1∑

j=0

[φ(Pj)

φ(P )

]r [ ‖χP‖Lp(·)(Rn)

‖χPj‖Lp(·)(Rn)

]r 1r

. (3.18)

On the other hand, by [82, Lemma 2.6], we find that

‖χPj‖Lp(·)(Rn) & 2

−j np+ 2

jPnp+ ‖χP‖Lp(·)(Rn)

and, by the condition (S1) of φ, we see that φ(P ) ≥ 2j log2 c12−jP log2 c1φ(cP , 2−j),

which, together with (3.18) and the condition (S2) of φ, implies that

IP,1. ‖f‖F

s(·),φp(·),q(·)

(Rn)

jP−1∑

j=0

2j( n

p+−log2 c1)r

[φ(cP , 2

−j)

φ(Pj)

]r 1r

2jP (log2 c1− n

p+)

. ‖f‖F

s(·),φp(·),q(·)

(Rn)

jP−1∑

j=0

2j( n

p+−log2 c1)r

1r

2jP (log2 c1− n

p+) ∼ ‖f‖

Fs(·),φp(·),q(·)

(Rn),

where we used the fact that c1 ∈ (0, 2n/p+) in the last inequality. Therefore,∥∥∥f∣∣∣F s(·),φ

p(·),q(·)(Rn)∥∥∥1= sup

P∈QIP . ‖f‖

Fs(·),φp(·),q(·)

(Rn),

which completes the proof of (i).Now we prove (ii). For all Q ∈ Q∗, from (3.17) and Theorem 3.11, we deduce

that, for all x ∈ Q,

[φ(Q)]−1‖χQ‖Lp(·)(Rn)|Q|−s(x)n |ϕjQ ∗ f(x)|

.‖χQ‖Lp(·)(Rn)

φ(Q)infy∈Q

|Q|− s(y)n (ϕ∗

jQf)a(y)

.1

φ(Q)

∥∥∥|Q|−s(·)n (ϕ∗

jQf)a

∥∥∥Lp(·)(Q)

. ‖f‖F

s(·),φp(·),q(·)

(Rn),

which implies that ‖f |F s(·),φp(·),q(·)(R

n)‖2 . ‖f‖F

s(·),φp(·),q(·)

(Rn).


Conversely, by choosing r ∈ (0,min1, p−, q−) and an argument similar tothat used in the proof of (i), we conclude that, for any P ∈ Q,

1

φ(P )

∥∥∥∥∥∥∥

∞∑

j=(jP∨0)

[2js(·)|ϕj ∗ f |

]q(·)

1q(·)

∥∥∥∥∥∥∥Lp(·)(P )

.∥∥∥f∣∣∣F s(·),φ

p(·),q(·)(Rn)∥∥∥2

∥∥∥∥∥∥∥∥∥∥

∞∑

j=(jP∨0)

∑

ℓ(Q)=2−j

Q∈Q,Q⊂P

φ(P )−1φ(Q)χQ‖χQ‖Lp(·)(Rn)

q(·)

1q(·)

∥∥∥∥∥∥∥∥∥∥Lp(·)(P )

.∥∥∥f∣∣∣F s(·),φ

p(·),q(·)(Rn)∥∥∥2

∞∑

j=(jP∨0)2j( n

p−+log2 c1)r

1r

2−jP ( n

p−+log2 c1)

.∥∥∥f∣∣∣F s(·),φ

p(·),q(·)(Rn)∥∥∥2,

where we used the fact that c1 ∈ (0, 2−n/p−) in the last inequality, which impliesthat

‖f‖F

s(·),φp(·),q(·)

(Rn). ‖f |F s(·),φ

p(·),q(·)(Rn)‖2.

This finishes the proof of (ii) and hence Theorem 3.12.

Remark 3.13. In the case that p, q, s and φ are as in Remark 1.5(ii), Theorem3.12(i) coincides with [80, Corollary 3.3(i)] and Theorem 3.12(ii) goes back to [78,Theorem 2.2(i)].

We now compare the Triebel-Lizorkin-type space with variable exponents in

this article with the variable Triebel-Lizorkin-Morrey space Es(·)p(·),q(·),u(Rn) intro-

duced by Ho [31] and show that, in general, these two scales of Triebel-Lizorkinspaces do not cover each other.

To recall the definition of the variable Triebel-Lizorkin-Morrey space in [31], weneed some notions. A measurable function u(x, r) : Rn× (0,∞) → (0,∞) is saidto belong to Wq with q ∈ (0,∞) if there exist C1, C2 ∈ (0,∞) and λ ∈ [0, 1/q)

such that, for all x ∈ Rn, u(x, r) > 1 if r ∈ [1,∞), u(x,2r)u(x,r)

≤ 4nλ if r ∈ (0,∞), and

C−12 ≤ u(x, t)

u(x, r)≤ C2 if 0 < r ≤ t ≤ 2r.

Definition 3.14. Let p, q, s, ϕjj∈Z+ be as in Definition 1.4 and u ∈ Wp+ .

Then the variable Triebel-Lizorkin-Morrey space Es(·)p(·),q(·),u(Rn) is defined to be

the set of all f ∈ S ′(Rn) such that

‖f‖Es(·)p(·),q(·),u

(Rn):= sup

z∈Rn

R∈(0,∞)

1

u(z, R)

∥∥∥∥∥∥

∞∑

j=0

[2js(·)|ϕj ∗ f |

]q(·) 1

q(·)

∥∥∥∥∥∥Lp(·)(B(z,R))

<∞.


Remark 3.15. (i) We point out that the Triebel-Lizorkin-type space with variableexponents in this article can not be covered by the Triebel-Lizorkin-Morrey spacein [31] even when c1 ∈ (0, 2n/p+). To see this, it suffices to show that there exists aset function φ satisfying (S1) and (S2) does not belong to Wq for any q ∈ (0,∞).

Indeed, for all Q ⊂ Rn, let φ(Q) :=∫Q|x|α dx, where α ∈ (−n, 0). Then, by

[65, p. 196], we know that φ is doubling, which, together with Remark 1.3(iv),further implies that φ satisfies the conditions (S1) and (S2). However, φ /∈ Wq

for any q ∈ (0,∞). To see this, let x0 ∈ Rn and r ∈ (1, 2) satisfy |x0| ≥ 2r. Then

φ(x0, r) := φ(Q(x0, r)) =

∫

Q(x0,r)

|y|α dy ∼ |x0|α

tends 0 as |x0| → ∞ since α ∈ (−n, 0), which implies that φ /∈ Wq for anyq ∈ (0,∞).

(ii) Also, the variable Triebel-Lizorkin-Morrey space investigated in [31] cannot be covered by the Triebel-Lizorkin-type space with variable exponents in thisarticle. To see this, it suffices to show that there exists a function u such that ubelongs to W1 but does not satisfy the condition (S2).

Indeed, let, for all x ∈ Rn and r ∈ (0,∞), u(x, r) := rλ(x), where λ(x) :=n(1 − 1

1+|x|). Then, as was pointed out in [31, p. 380], u ∈ W1. However, u

does not satisfy the condition (S2). To see this, let x, y ∈ Rn satisfy that

ε < |x| < (1+ε)r+ε2+ε

and |y| = 1+|x|1+ε

− 1, where ε ∈ (0,∞) and r ∈ (ε,∞). Then

|x− y| ≤ |x|+ |y| = |x|+ 1 + |x|1 + ε

− 1 < r,

butu(x, r)

u(y, r)= r

n( 11+|y|

− 11+|x|

)= r

nε1+|x| → ∞, as r → ∞,

which implies that u does not satisfy the condition (S2).

As an application of Theorem 3.12, we prove that the space F 0,φp(·),2(R

n) coincides

with the Morrey space with variable exponent, Mp(·)φ (Rn), which is defined to be

the set of all measurable functions f such that

‖f‖Mp(·)φ (Rn)

:= supP∈Q

1

φ(P )‖f‖Lp(·)(P ) <∞,

where the supremum is taken over all dyadic cubes of Rn.

Remark 3.16. (i) We point out that, in [31], Ho studied the variable Morrey

space Mp(·)u (Rn), which is defined in the same way as Mp(·)

φ (Rn) above but withφ replaced by u as in Definition 3.14 and the supremum is taken over all balls ofRn. From Remark 3.15, we deduce that the Morrey space with variable exponent

Mp(·)φ (Rn) in this article and the variable Morrey space Mp(·)

u (Rn) in [31] do notcover each other.

(ii) For ϕ : Rn × (0,∞) → (0,∞) and a variable exponent p : Rn → [1,∞),Nakai [52] introduced the variable Morrey space L(p,ϕ)(Rn), which is defined to


be the set of all measurable functions f such that

‖f‖L(p,ϕ)(Rn) := supballs B⊂Rn

‖f‖p,ϕ,B <∞,

where, for all balls B := B(x, r) ⊂ Rn, ϕ(B) := ϕ(x, r) and

‖f‖p,ϕ,B := inf

λ ∈ (0,∞) :

1

ϕ(B)|B|

∫

B

[ |f(y)|λ

]p(y)dy ≤ 1

,

and the supremum is taken over all balls B of Rn.We claim that, if there exists a positive constant C such that, for all x ∈ Rn

and 0 < r < s <∞,

C−1φ(x, r) ≤ φ(x, s) ≤ Cφ(x, r) (3.19)

and, for all balls B ⊂ Rn and all y ∈ B,

ϕ(B)|B| ∼ [φ(B)]p(y), (3.20)

then Mp(·)φ (Rn) coincides with L(p,ϕ)(Rn).

Indeed, by (3.19) and the definition of ‖ · ‖Mp(·)φ (Rn)

, we conclude that

‖f‖Mp(·)φ (Rn)

∼ supballs B⊂Rn

inf

λ ∈ (0,∞) :

∫

B

[ |f(y)|φ(B)λ

]p(y)dy ≤ 1

. (3.21)

On the other hand, by (3.20), we find that

inf

λ ∈ (0,∞) :

∫

B

[ |f(y)|φ(B)λ

]p(y)dy ≤ 1

∼ inf

λ ∈ (0,∞) :

1

ϕ(B)|B|

∫

B

[ |f(y)|λ

]p(y)dy ≤ 1

which, combined with (3.21), implies that Mp(·)φ (Rn) coincides with L(p,ϕ)(Rn).

This proves the above claim.Obviously, in general, these two scales of Morrey spaces with variable expo-

nents, Mp(·)φ (Rn) and L(p,ϕ)(Rn), may not cover each other.

In what follows, for all p ∈ P(Rn), denote by Lp(·)(ℓ2(Rn)) the set of all se-quences gjj∈Z+ of measurable functions such that

‖gjj∈Z+‖Lp(·)(ℓ2(Rn)) :=

∥∥∥∥∥∥∥

∑

j∈Z+

|gj|2

12

∥∥∥∥∥∥∥Lp(·)(Rn)

<∞.

Let (ϕ,Φ) and (ψ,Ψ) be two pairs of admissible functions satisfying (2.1). Theoperator G is defined by setting, for all f ∈ Lp(·)(Rn), G(f) := ϕj ∗ fj∈Z+,where, when j = 0, ϕ0 is replaced by Φ, and its conjugate operator G∗ is defined

by setting, for all gjj∈Z+ ∈ LP (·)(ℓ2(Rn)),

G∗(gjj∈Z+) :=∑

j∈Z+

ψj ∗ gj,


where, when j = 0, ψ0 is replaced by Ψ.

Remark 3.17. Let p(·) ∈ C log(Rn) satisfy 1 < p− ≤ p+ <∞. Then, from the fact

that LP (·)(Rn) = F 0P (·),2(R

n) (see [20, Theorem 4.2]), we deduce that the operator

G is bounded from LP (·)(Rn) to LP (·)(ℓ2(Rn)). Furthermore, by an argumentsimilar to that used in the proof of [31, Corollary 4.4], we conclude that theoperator G∗ is bounded from Lp(·)(ℓ2(Rn)) to Lp(·)(Rn).

Proposition 3.18. Let p and φ be as in Definition 1.4 and c1 ∈ (0, 2n/p+). If1 < p− ≤ p+ <∞, then

Mp(·)φ (Rn) = F 0,φ

p(·),2(Rn)

with equivalent norms.

Proof. We first prove that Mp(·)φ (Rn) → F 0,φ

p(·),2(Rn). By Theorem 3.12(i), it suf-

fices to show that, for all f ∈ Mp(·)φ (Rn),

supQ∈Q

1

φ(Q)

∥∥∥∥∥∥

∞∑

j=0

|ϕj ∗ f |2 1

2

∥∥∥∥∥∥Lp(·)(Q)

. ‖f‖Mp(·)φ (Rn)

, (3.22)

where ϕj∞j=0 are as in Definition 1.4.For all Q := Q(x0, r) ∈ Q, let f1 := fχQ(x0,2r) and f2 := f − f1. From [20,

Theorem 4.2] and the condition (S1) of φ, we deduce that

I1 :=1

φ(Q)

∥∥∥∥∥∥

∞∑

j=0

|ϕj ∗ f1|2 1

2

∥∥∥∥∥∥Lp(·)(Q)

.1

φ(Q)‖f1‖Lp(·)(Rn) ∼

1

φ(Q)‖f‖Lp(·)(Q(x0,2r)) . ‖f‖Mp(·)

φ (Rn). (3.23)

On the other hand, by the Minkowski inequality, we find that, for all x ∈ Rn,

∞∑

j=0

|ϕj ∗ f2(x)|2 1

2

.

∫

Rn\Q(x0,2r)

∞∑

j=0

|ϕj(x− y)|2 1

2

|f(y)| dy

.

∫

Rn\Q(x0,2r)

|f(y)||x− y|n dy ∼

∞∑

k=1

∫

Sk

|f(y)||x− y|n dy,

where, for k ∈ N, Sk := Qk+1 \ Qk and Qk := Q(x0, 2kr). Observe that, when

x ∈ Q(x0, r) and y ∈ Sk, |x − y| ≥ 2kr. Setting (p(·))∗ := p(·)p(·)−1

, by the Holder

inequality of variable Lebesgue spaces (see Remark 1.1(iii)), [82, Lemma 2.6] and[31, Proposition 2.4], we see that

∥∥∥∥∥∥

∞∑

j=0

|ϕj ∗ f2|2 1

2

∥∥∥∥∥∥Lp(·)(Q)


.

∞∑

k=1

1

2knrn‖f‖Lp(·)(Sk)

‖χSk‖Lp(·)∗(Rn)‖χQ‖Lp(·)(Rn)

.

∞∑

k=1

2−kn/p+

rn2kn‖f‖Lp(·)(Sk)

‖χQk+1‖Lp(·)∗(Rn)‖χQk+1

‖Lp(·)(Rn)

.

∞∑

k=1

2−kn/p+‖f‖Lp(·)(Sk),

which, together with the condition (S1) of φ, implies that

I2 :=1

φ(Q)

∥∥∥∥∥∥

∞∑

j=0

|ϕj ∗ f2|2 1

2

∥∥∥∥∥∥Lp(·)(Q)

.

∞∑

k=1

2−kn/p+‖f‖Lp(·)(Qk+1)

φ(Qk+1)

φ(Qk+1)

φ(Q)

. ‖f‖Mp(·)φ (Rn)

∞∑

k=1

2−kn/p+2k log2 c1 ∼ ‖f‖Mp(·)φ (Rn)

, (3.24)

where we used the fact that c1 ∈ (0, 2n/p+) in the last inequality.Combining (3.23) and (3.24), we conclude that

supQ∈Q

1

φ(Q)

∥∥∥∥∥∥

∞∑

j=0

|ϕj ∗ f |2 1

2

∥∥∥∥∥∥Lp(·)(Q)

. supQ∈Q

(I1 + I2) . ‖f‖Mp(·)φ (Rn)

and (3.22) holds true.

Next, we prove that F 0,φp(·),2(R

n) → Mp(·)φ (Rn). Let f ∈ F 0,φ

p(·),2(Rn). Then, by

the Calderon reproducing formula (see [80, Lemma 2.3]), we find that

f = Ψ ∗ Φ ∗ f +∞∑

j=1

ψj ∗ ϕj ∗ f =:∞∑

j=0

ψj ∗ ϕj ∗ f (3.25)

in S ′(Rn), where Ψ, Φ, ϕ and ψ are as in (2.1). For all j ∈ Z+, we use fj todenote ϕj ∗ f . For all Q := Q(x0, r) ∈ Q and j ∈ Z+, let f

1j := fjχQ(x0,2r) and

f 2j := fj − f 1

j . Then we know that

1

φ(Q)

∥∥∥∥∥

∞∑

j=0

ψj ∗ ϕj ∗ f∥∥∥∥∥Lp(·)(Q)

.1

φ(Q)

∥∥∥∥∥

∞∑

j=0

ψj ∗ f 1j

∥∥∥∥∥Lp(·)(Q)

+1

φ(Q)

∥∥∥∥∥

∞∑

j=0

ψj ∗ f 2j

∥∥∥∥∥Lp(·)(Q)

=: J1 + J2.

By Remark 3.17, the condition (S1) of φ and Theorem 3.12(i), we see that

J1∼1

φ(Q)

∥∥∥G∗(f 1j

j∈Z+

)∥∥∥Lp(·)(Rn)

.1

φ(Q)

∥∥∥∥∥∥

∞∑

j=0

|ϕj ∗ f |2 1

2

∥∥∥∥∥∥Lp(·)(Q(x0,2r))

. ‖f‖F 0,φp(·),2

(Rn). (3.26)


On the other hand, for all x ∈ Q(x0, 2r), by the Holder inequality, we find that∣∣∣∣∣∣

∑

j∈Z+

ψj ∗ f 2j (x)

∣∣∣∣∣∣.

∫

Rn\Q(x0,2r)

∑

j∈Z+

|fj(y)|2

12

|x− y|−n dy.

Thus, by an argument similar to that used in the proof of (3.24), we concludethat

J2 . ‖f‖F 0,φp(·),2

(Rn),

which, combined with (3.26), implies that

1

φ(Q)

∥∥∥∥∥

∞∑

j=0

ψj ∗ ϕj ∗ f∥∥∥∥∥Lp(·)(Q)

. J1 + J2 . ‖f‖F 0,φp(·),2

(Rn).

Therefore,∑∞

j=0 ψj ∗ ϕj ∗ f ∈ Mp(·)φ (Rn) and

∥∥∥∥∥

∞∑

j=0

ψj ∗ ϕj ∗ f∥∥∥∥∥Mp(·)

φ (Rn)

. ‖f‖F 0,φp(·),2

(Rn),

which, together with (3.25), implies that F 0,φp(·),2(R

n) → Mp(·)φ (Rn). This finishes

the proof of Proposition 3.18.

Remark 3.19. In the case that p and φ are as in Remark 1.5(ii), the conclusionof Proposition 3.18 is already known; see, for example, [62, Theorem 3.9].

We end this section by giving another application of Theorem 3.11.

Proposition 3.20. Let p, q, s and φ be as in Definition 1.4. Then

S(Rn) → Fs(·),φp(·),q(·)(R

n) → S ′(Rn).

Proof. We first prove that S(Rn) → Fs(·),φp(·),q(·)(R

n). To prove this embedding, we

need to show that there exists an M ∈ N such that, for all f ∈ S(Rn),

‖f‖F

s(·),φp(·),q(·)

(Rn). ‖f‖SM (Rn).

Let f ∈ S(Rn) and (ϕ,Φ) be a pair of admissible functions. Let P := QjP kP

be an arbitrary dyadic cube. If jP > 0, choosing r ∈ (0,min1, p−, q−), by [80,Lemma 2.4], Remark 1.1(i) and Lemmas 2.6 and 2.7, we obtain

1

φ(P )

∥∥∥∥∥∥

∞∑

j=jP

[2js(·)|ϕj ∗ f |

]q(·) 1

q(·)

∥∥∥∥∥∥Lp(·)(P )

. ‖f‖SM+1(Rn)1

φ(P )

∥∥∥∥∥∥

∞∑

j=jP

[2js(·)

2−jM

(1 + | · |)n+M]q(·) 1

q(·)

∥∥∥∥∥∥Lp(·)(P )

. ‖f‖SM+1(Rn)1

φ(P )

∞∑

j=jP

2−j(M−s+)r

∥∥∥∥1

(1 + | · |)(M+n)r

∥∥∥∥L

p(·)r (P )

1r


. ‖f‖SM+1(Rn)

∞∑

j=jP

2−j(M−s+)r 1

(1 + 2−jP |kP |)(M+n)r

1r ‖χP‖Lp(·)(P )

φ(P )

. ‖f‖SM+1(Rn)2−jP (M

2+ n

p+−s+−log2 c1)(1 + |kP |)−

M2+n( 1

p−− 1

p+)+log2(c1c1)

. ‖f‖SM+1(Rn), (3.27)

where M is chosen large enough.If jP ≤ 0, then we see that

IP :=1

φ(P )

∥∥∥∥∥∥

∞∑

j=0

[2js(·)|ϕj ∗ f |

]q(·) 1

q(·)

∥∥∥∥∥∥Lp(·)(P )

.1

φ(P )‖Φ ∗ f‖Lp(·)(P ) +

1

φ(P )

∞∑

j=1

2js+r‖ϕj ∗ f‖rLp(·)(P )

1r

.

When P is away from the origin, by an argument similar to that used in theproof of (3.27), we conclude that IP . ‖f‖SM+1(Rn) with M being sufficiently

large. When one of the corners of P is the origin, then P ⊂ ∪−jP+ni=0 Si, where

S0 := B(0, 1) and Si := 2iS0\(2i−1S0) for all i ∈ 1, . . . ,−jP + 1. From this,Lemmas 2.6 and 2.7 and the fact that |kP | ≤ 1, we deduce that

1

φ(P )‖Φ ∗ f‖Lp(·)(P ).

1

φ(P )

−jP+n∑

i=0

∥∥∥∥1

(1 + | · |)M∥∥∥∥r

Lp(·)(Si)

1r

. ‖f‖SM+1(Rn)

and, similarly,

1

φ(P )

∞∑

j=1

2js+r‖ϕj ∗ f‖rLp(·)(P )

1r

. ‖f‖SM+1(Rn),

whereM is chosen large enough, which implies that IP . ‖f‖SM+1(Rn). Therefore,

S(Rn) → Fs(·),φp(·),q(·)(R

n) and ‖f‖F

s(·),φp(·),q(·)

(Rn). ‖f‖SM+1(Rn).

Next we show that Fs(·),φp(·),q(·)(R

n) → S ′(Rn). To this end, we need to prove that

there exists an M ∈ N such that, for all f ∈ Fs(·),φp(·),q(·)(R

n) and h ∈ S(Rn),

|〈f, h〉| . ‖f‖F

s(·),φp(·),q(·)

(Rn)‖h‖SM+1(Rn).

Let ϕ, ψ, Φ and Ψ be as in Theorem 2.3. Then, by the Calderon reproducingformula in [80, Lemma 2.3], together with [80, Lemma 2.4], we obtain

|〈f, h〉|≤∫

Rn

|Φ ∗ f(x)||Ψ ∗ h(x)| dx+∞∑

j=1

∫

Rn

|ϕj ∗ f(x)||ψj ∗ h(x)| dx

. ‖h‖SM+1(Rn)

∞∑

j=0

2−jM∫

Rn

|ϕj ∗ f(x)|(1 + |x|)−(n+M) dx


∼‖h‖SM+1(Rn)

∞∑

j=0

2−jM∑

k∈Zn

∫

Q0k

|ϕj ∗ f(x)|(1 + |x|)−(n+M) dx, (3.28)

where we used Φ to replace ϕ0. Notice that, for any j ∈ Z+, k ∈ Zn, a ∈ (0,∞)and y ∈ Qjk,∫

Q0k

|ϕj ∗ f(x)| dx. (ϕ∗jf)a(y)

∫

Q0k

(1 + 2j|x|+ 2j |y|)a dx

. 2ja(ϕ∗jf)a(y)(1 + |k|)a.

Then, by the arbitrariness of y ∈ Qjk, we see that∫

Q0k

|ϕj ∗ f(x)| dx . 2ja(1 + |k|)a infy∈Qjk

(ϕ∗jf)a(y),

which, combined with (3.28), Theorem 3.11 and Lemmas 2.6 and 2.7, implies that

|〈f, h〉|. ‖h‖SM+1(Rn)

∞∑

j=0

2−jM∑

k∈Zn

∫

Q0k

|ϕj ∗ f(x)|(1 + |k|)n+M dx

. ‖h‖SM+1(Rn)

∞∑

j=0

2−jM+ja∑

k∈Zn

infy∈Qjk(ϕ∗

jf)a(y)

(1 + |k|)n+M−a

. ‖h‖SM+1(Rn)

∞∑

j=0

2−jM+ja∑

k∈Zn

(1 + |k|)−(n+M−a)‖(ϕ∗jf)a‖Lp(·)(Qjk)

‖χQjk‖Lp(·)(Qjk)

. ‖f‖F

s(·),φp(·),q(·)

(Rn)‖h‖SM+1(Rn)

∞∑

j=0

2j(a−M−s−)

×∑

k∈Zn

(1 + |k|)(a−n−M) φ(Qjk)

‖χQjk‖Lp(·)(Qjk)

. ‖f‖F

s(·),φp(·),q(·)

(Rn)‖h‖SM+1(Rn),

where a is chosen as in (3.12). This finishes the proof of Proposition 3.20.

4. A trace theorem

In this section, we mainly establish a trace theorem for Triebel-Lizorkin-typespaces with variable exponents by applying the atomic characterization of thesespaces obtained in Theorem 3.8.

To state our main result of this section, we first give some notation. Formeasurable functions p, q, s and a set function φ being as in Definition 1.4, let

Fs(·,0),φp(·,0),q(·,0)(R

n−1) denote the Triebel-Lizorkin-type spaces with variable exponents

p(·, 0), q(·, 0) and s(·, 0) on Rn−1×0, where φ is defined by setting, for all cubes

Q of Rn−1, φ(Q) := φ(Q × [0, ℓ(Q)). In what follows, let Rn+ := Rn−1 × [0,∞)

and Rn− := Rn−1 × (−∞, 0].

Let f ∈ Fs(·),φp(·),q(·)(R

n). Then, by Theorem 3.8, we have f =∑

Q∈Q∗ tQaQ in

S ′(Rn) and

‖tQQ∈Q∗‖fs(·),φp(·),q(·)

(Rn)≤ C‖f‖

Fs(·),φp(·),q(·)

(Rn),


where C is a positive constant independent of f and, for each Q ∈ Q∗, aQ is a

smooth atom of Fs(·),φp(·),q(·)(R

n). Define the trace of f by setting, for all x ∈ Rn−1,

Tr(f)(x) :=∑

Q∈Q∗

tQaQ(x, 0). (4.1)

This definition of Tr(f) is determined canonical for all f ∈ Fs(·),φp(·),q(·)(R

n), since the

actual construction of aQ in the proof of Theorem 3.8 implies that tQaQ is obtainedcanonically. Moreover, in Lemma 4.3 below, we show that the summation in (4.1)converges in S ′(Rn−1). Thus, the trace operator is well defined.

The main result of this section is the following trace theorem.

Theorem 4.1. Let n ≥ 2, p, q ∈ P(Rn) satisfy

0 < p− ≤ p+ <∞, 0 < q− ≤ q+ <∞and 1

p, 1q∈ C log(Rn), s ∈ C log

loc (Rn) ∩ L∞(Rn) and φ be a set function satisfying

the conditions (S1) and (S2). If

s− − 1

p−− (n− 1)

[1

min1, p−− 1

]> 0, (4.2)

then

TrFs(·),φp(·),q(·)(R

n) = Fs(·,0)− 1

p(·,0),φ

p(·,0),p(·,0) (Rn−1).

Remark 4.2. (i) When p, q, s and φ are as in Remark 1.5(ii), Theorem 4.1goes back to [80, Theorem 6.8]. Moreover, Theorem 4.1 coincides with the tracetheorem for the classical Triebel-Lizorkin space F s

p,q(Rn) with constant exponents

(see [27, Theorem 11.1 and p. 134]) and, in this case, the condition (4.2) is optimal.(ii) In the case that φ is as in Remark 1.5(i), it was proved in [20, Theorem

3.13] (see also [56, Theorem 5.1(1)]) that the conclusion of Theorem 4.1 is true ifs and p satisfy that, for all x ∈ Rn,

s(x)− 1

p(x)− (n− 1)

[1

min1, p(x) − 1

]> δ (4.3)

for some δ ∈ (0,∞), which is a little weaker than (4.2). The reason that theassumption (4.2), in this case, is a little stronger than that in [20, Theorem 3.13](see also (4.3)) comes from an application of Theorem 3.8, which, in this case,can be further refined; see Remark 3.9(ii).

To prove Theorem 4.1, we first need to show that (4.1) converges in S ′(Rn−1).

Lemma 4.3. Let p, q, s and φ be as in Theorem 4.1 satisfying (4.2). Then, for

all f ∈ Fs(·),φp(·),q(·)(R

n), Tr(f) ∈ S ′(Rn−1).

Proof. Let f ∈ Fs(·),φp(·),q(·)(R

n). Then, by Theorem 3.8, we can write

f =∑

Q∈Q∗

tQaQ


in S ′(Rn) and‖tQQ∈Q∗‖

fs(·),φp(·),q(·)

(Rn). ‖f‖

Fs(·),φp(·),q(·)

(Rn),

where, for each Q ∈ Q∗, aQ is a (K, L)-smooth atom supported near Q of

Fs(·),φp(·),q(·)(R

n) with K ∈ (s+ + log2 c1,∞) and L is as in (3.1). Let

A :=Q ∈ Q∗ : 3Q ∩ (x, xn) ∈ Rn−1 × R : xn = 0 6= ∅

,

where Q denotes the closure of Q in Rn. Since supp aQ ⊂ 3Q for all Q ∈ Q∗, itfollows that aQ(·, 0) = 0 if Q /∈ A. Observe that, if Qjk ∈ A with j ∈ Z+ andk := (k1, . . . , kn) ∈ Zn, then |kn| ≤ 2. Therefore,

∞∑

j=0

∑

k∈Zn

tQjkaQjk

(·, 0) =∞∑

j=0

∑

k∈Zn, |kn|≤2

tQjkaQjk

(·, 0).

Thus, to complete the proof of Lemma 4.3, it suffices to show that

limN→∞,Λ→∞

N∑

j=0

∑

k∈Zn, |kn|≤2|k|≤Λ

tQjkaQjk

(·, 0) (4.4)

exists in S ′(Rn−1). By (4.2), we see that

s− − n

p−+

min1, p−p−

(n− 1)

= s− − 1

p−− n− 1

p−(1−min1, p−)

≥ s− − 1

p−− n− 1

min1, p−(1−min1, p−)

= s− − 1

p−− (n− 1)

[1

min1, p−− 1

]> 0,

which implies that there exists r ∈ (0,min1, p−) such that

s− − n

p−+

r

p−(n− 1) > 0.

Let p(·) and s(·) be as in the proof of Theorem 3.8. Then s− − rp−

> 0. For all

j ∈ Z+ and k ∈ Zn, let

∆k,j := Rn−1 × [kn2−j, (kn + 1)2−j).

Then, by the smoothness condition (A3), we know that, for all h ∈ S(Rn−1) andj ∈ Z+,

I :=

∣∣∣∣∣∣∣

⟨∑


tQjkaQjk

(·, 0), h(·)⟩∣∣∣∣∣∣∣

. 2jn/2∑


|tQjk|∫

Rn−1

(1 + |y|)−δ(1 + 2j |(y, 0)− xQjk

|)R dy


∼ 2jn/2+j∑


|tQjk|∫

∆k,j

(1 + |y|)−δ(1 + 2j|(y, 0)− xQjk

|)R dydyn

. 2jn/2+j∑


|tQjk|∫

Rn

(1 + |y|)−δχ∆k,j(y)

(1 + 2j|y − xQjk|)R dy,

where R ∈ (0,∞) is chosen large enough, and δ ∈ (0,∞) will be determined later.By an argument similar to that used in the proof of Theorem 3.8, we find that

I . 2−j(s−−1)‖t‖fs(·),φp(·),q(·)

(Rn)

∥∥∥∥χ∆j

(1 + | · |)δ−δ0

∥∥∥∥L(p(·))∗(Rn)

, (4.5)

where ∆j := Rn−1 × [−2−j+1, 2−j+1]. On the other hand, by choosing δ largeenough, we see that

∫

Rn

[χ∆j

(y)/(1 + |y|)−δ+δ02−j/(p(·)∗)+

](p(y))∗dy

. 2j∫

Rn−1×[−2−j+1,2−j+1]

[1

(1 + |y|)δ−δ0](p(·)∗)−

dydyn

.

∫

Rn−1

[1

(1 + |y|)δ−δ0](p(·)∗)−

dy . 1,

which, together with Remark 1.1(ii), implies that∥∥∥∥

χ∆j

(1 + | · |)δ−δ0

∥∥∥∥L(p(·))∗(Rn)

. 2−j/(p(·)∗)+ ∼ 2

−j(1− rp−

).

From this and (4.5), we deduce that I . 2−j(s−− r

p−)‖t‖

fs(·),φp(·),q(·)

(Rn), which, combined

with the fact that s− − rp−

> 0, implies that (4.4) converges in S ′(Rn−1). This

finishes the proof of Lemma 4.3.

Next we prove Theorem 4.1 by beginning with several technical lemmas.

Lemma 4.4. Let pi, qi, si and φ be as in Definition 1.4 with p, q, s replaced bypi, qi, si, respectively, where i ∈ 1, 2. If s1 ≤ s2, p1 ≤ p2, (p1)∞ = (p2)∞ andq1 ≥ q2, then

Fs2(·),φp2(·),q2(·)(R

n) → Fs1(·),φp1(·),q1(·)(R

n).

Proof. By [20, Proposition 6.5], we find that

Lp2(·)(Rn) → Lp1(·)(Rn).

From this, s1 ≤ s2 and (2.6), we can easily deduce the desired conclusion, thedetails being omitted. This finishes the proof of Lemma 4.4.

Lemma 4.5. Let pi, qi, si and φ be as in Theorem 4.1 with p, q, s replaced bypi, qi, si, where i ∈ 1, 2. Assume that s1 = s2 and p1 = p2 on Rn

+ or Rn−, and


that s1 ≤ s2 and p1 ≤ p2. If

(s2)− − 1

(p2)−− (n− 1)

[1

min1, (p2)− − 1

]> 0, (4.6)

then

TrFs1(·),φp1(·),q1(·)(R

n) = TrFs2(·),φp2(·),q2(·)(R

n);

moreover, if q(·) is as in Theorem 4.1, then

TrFs1(·),φp1(·),q1(·)(R

n) = TrFs1(·),φp1(·),q(·)(R

n).

To prove Lemma 4.5, we need the following conclusion.

Proposition 4.6. Let p, q, s, φ be as in Definition 1.4 and δ ∈ (0, 1). Supposethat, for each Q ∈ Q∗, EQ ⊂ 3Q is a measurable set with |EQ| ≥ δ|Q|. Then, forall t := tQQ∈Q∗ ⊂ C, t ∈ f

s(·),φp(·),q(·)(R

n) if and only if ‖t‖ ˜fs(·),φp(·),q(·)

(Rn)<∞, where

‖t‖ ˜fs(·),φp(·),q(·)

(Rn)

:= supP∈Q

1

φ(P )

∥∥∥∥∥∥∥

∞∑

j=(jP∨0)

∑

ℓ(Q)=2−j

[2js(·)|tQ||Q|−

12χEQ

]q(·)

1q(·)

∥∥∥∥∥∥∥Lp(·)(P )

.

Proof. We first suppose that ‖t‖ ˜fs(·),φp(·),q(·)

(Rn)< ∞ and show that t ∈ f

s(·),φp(·),q(·)(R

n).

Notice that, for all m ∈ (n,∞), Q ∈ Q∗ and x ∈ Q,

χQ(x) . ηjQ,m+Clog(s) ∗ χEQ(x).

From this, Lemmas 2.9 and 2.10, we deduce that

‖t‖fs(·),φp(·),q(·)

(Rn)

= supP∈Q

1

φ(P )

∥∥∥∥∥∥∥

∞∑

j=(jP∨0)

∑

ℓ(Q)=2−j

[2jrs(·)|tQ|r|Q|−

r2χQ

] q(·)r

1q(·)

∥∥∥∥∥∥∥Lp(·)(P )

. supP∈Q

1

φ(P )

∥∥∥∥∥∥∥∥∥

∞∑

j=(jP∨0)

ηj,m ∗

2jrs(·)

∑

Q⊂P

ℓ(Q)=2−j

|tQ|rχEQ

|Q| r2

q(·)r

1q(·)

∥∥∥∥∥∥∥∥∥Lp(·)(P )

. supP∈Q

1

φ(P )

∥∥∥∥∥∥∥∥∥

∞∑

j=(jP∨0)

2js(·)

∑

Q⊂P

ℓ(Q)=2−j

|tQ||Q|−12χEQ

q(·)

1q(·)

∥∥∥∥∥∥∥∥∥Lp(·)(P )

∼ ‖t‖ ˜fs(·),φp(·),q(·)

(Rn),


which implies that t ∈ fs(·),φp(·),q(·)(R

n).

Conversely, by an argument similar to the above and the fact that, for allm ∈ (n,∞), Q ∈ Q∗ and x ∈ EQ, χEQ

(x) . ηjQ,m+Clog(s)∗ χQ(x), we conclude

that, for all t ∈ fs(·),φp(·),q(·)(R

n), ‖t‖ ˜fs(·),φp(·),q(·)

(Rn). ‖t‖

fs(·),φp(·),q(·)

(Rn). This finishes the proof

of Proposition 4.6.

Proof of Lemma 4.5. From Remark 1.2(i) and the condition that p1 = p2 onRn

+ or Rn−, we deduce that (p1)∞ = (p2)∞. Let r0 := min(q2)−, (q1)− and

r1 := max(q2)+, (q1)+. Then, by Lemma 4.4 and (2.6), we see that

Fs2(·),φp2(·),r0(R

n) → Fs1(·),φp1(·),q1(·)(R

n) → Fs1(·),φp1(·),r1(R

n) (4.7)

andFs2(·),φp2(·),r0(R

n) → Fs2(·),φp2(·),q2(·)(R

n) → Fs1(·),φp1(·),r1(R

n). (4.8)

By Lemma 4.3 and an argument similar to that used in the proof of [20, Lemma7.2], with [20, Theorem 3.8 and Lemma 7.1] replaced by Theorem 3.8 and Propo-

sition 4.6, we conclude that, for all f ∈ Fs1(·),φp1(·),r1(R

n), Tr(f) exists in S ′(Rn−1) and

TrFs1(·),φp1(·),r1(R

n) ⊂ TrFs2(·),φp2(·),r0(R

n). From this, (4.7) and (4.8), we deduce that

TrFs1(·),φp1(·),q1(·)(R

n) ⊂ TrFs1(·),φp1(·),r1(R

n) ⊂ TrFs2(·),φp2(·),r0(R

n) ⊂ TrFs2(·),φp2(·),q2(·)(R

n)

⊂ TrFs1(·),φp1(·),r1(R

n) ⊂ TrFs2(·),φp2(·),r0(R

n) ⊂ TrFs1(·),φp1(·),q1(·)(R

n),


Remark 4.7. By the proof of Lemma 4.5, we see that the condition (4.6) is only

used to ensure that TrFs2(·),φp2(·),q2(·)(R

n) exists in S ′(Rn−1). Thus, by an argument

similar to that used in the proof of Lemma 4.5, we have the following conclusion,the details being omitted. Under the same assumption as in Lemma 4.5, if, for

all f ∈ Fs2(·),φp2(·),q2(·)(R

n), the trace of f defined as in (4.1) exists in S ′(Rn−1), then,

for all g ∈ Fs1(·),φp1(·),q1(·)(R

n), the trace of g defined as in (4.1) also exists in S ′(Rn−1);moreover,

TrFs1(·),φp1(·),q1(·)(R

n) = TrFs2(·),φp2(·),q2(·)(R

n).

Lemma 4.8. Let pi, q, si be as in Theorem 4.1 with p and s replaced by pi and si,i ∈ 1, 2. Assume that s1(x) = s2(x) and p1(x) = p2(x) for all x ∈ Rn−1 × 0.If (4.2) is satisfied with (s, p) replaced, respectively, by (s1, p1) and (s2, p2), then

TrFs1(·),φp1(·),q(·)(R

n) = TrFs2(·),φp2(·),p2(·)(R

n).

Proof. For all x ∈ Rn and i ∈ 1, 2, let si(x) := si(x) if x ∈ Rn− and si(x) :=

mins1(x), s2(x) otherwise, and, for all x ∈ Rn, s(x) := mins1(x), s2(x). Sim-ilarly, for i ∈ 1, 2, let pi(x) := pi(x) if x ∈ Rn

− and

pi(x) := minp1(x), p2(x)otherwise, and, for all x ∈ Rn,

p(x) := minp1(x), p2(x).


Then, by applying Lemma 4.5 and Remark 4.7, we conclude that

TrFs1(·),φp1(·),q(·)(R

n) =TrFs1(·),φp1(·),p1(·)(R

n) = TrFs1(·),φp1(·),p1(·)(R

n)

=TrFs(·),φp(·),p(·)(R

n) = TrFs2(·),φp2(·),p2(·)(R

n) = TrFs2(·),φp2(·),p2(·)(R

n),


In what follows, let Q(Rn) := Q and Q∗(Rn) := Q∗. Denote by Q(Rn−1) the

set of all dyadic cubes of Rn−1 and Q∗(Rn−1) the set of all dyadic cubes Q of

Rn−1 with ℓ(Q) ≤ 1.

Proof of Theorem 4.1. By Lemma 4.8, we may assume that q = p with p ands independent of the n-th coordinate xn with |xn| ≤ 2. Indeed, let, for all(x, xn) ∈ Rn−1 × [−2, 2], p0(x, xn) := p(x, 0). Then p0 ∈ C log(Rn−1 × [−2, 2]).By [19, Proposition 4.1.7], we find that p0 has an extension p ∈ C log(Rn) withp− = (p0)− and p∞ = (p0)∞. Define s by setting, for all (x, xn) ∈ Rn−1 × R,

s(x, xn) := s(x, 0). Then it is easy to see that s ∈ C logloc (R

n)∩L∞(Rn). Moreover,p and s are independent of the n-th coordinate xn with |xn| ≤ 2, and satisfy

s− − 1

p−− (n− 1)

[1

min1, p−− 1

]> 0.

Then, by Lemma 4.8, we see that

TrFs(·),φp(·),q(·)(R

n) = TrFs(·),φp(·),p(·)(R

n).

For notational simplicity, let, for all x ∈ Rn−1,

β(x, 0) := s(x, 0)− 1

p(x, 0),

Fβ(·,0),φp(·,0) (Rn−1) := F

β(·,0),φp(·,0),p(·,0)(R

n−1)

and

fβ(·,0),φp(·,0) (Rn−1) := f

β(·,0),φp(·,0),p(·,0)(R

n−1).

We finish the proof of Theorem 4.1 by two steps.

Step 1) We show that, for all f ∈ Fs(·),φp(·),q(·)(R

n), Tr(f) ∈ S ′(Rn−1) and

‖Tr(f)‖F

β(·,0),φp(·,0)

(Rn−1). ‖f‖

Fs(·),φp(·),p(·)

(Rn). (4.9)

Without loss of generality, we may assume that ‖f‖F

s(·),φp(·),p(·)

(Rn)= 1. By Theorem

3.8, we see that

f =∑

Q∈Q∗(Rn)

tQaQ

in S ′(Rn), where, for all Q ∈ Q∗, aQ is a (K, L)-smooth atom supported near Q

of Fs(·),φp(·),p(·)(R

n) with

K ∈ (s+ +max0, log2 c1,∞) and L ∈(

n

min1, p−− n− s−,∞

)(4.10)


and t := tQQ∈Q∗(Rn) ∈ fs(·),φp(·),p(·)(R

n), which can be chosen such that

‖t‖fs(·),φp(·),p(·)

(Rn). ‖f‖

Fs(·),φp(·),p(·)

(Rn). (4.11)

Since supp aQ ⊂ 3Q, it follows that, if i /∈ 0, 1, 2, thenaQ×[(i−1)ℓ(Q),iℓ(Q))(·, 0) = 0,

which implies that Tr(f) can be rewritten as

2∑

i=0

∑

Q∈Q∗(Rn−1)

tQ×[(i−1)ℓ(Q),iℓ(Q))aQ×[(i−1)ℓ(Q),iℓ(Q))(·, 0).

Therefore, to show (4.9), by Theorem 3.8 again, it can be reduced to prove thateach

b(i)

Q:= [ℓ(Q)]

12aQ×[(i−1)ℓ(Q),iℓ(Q))

is a (K, L)-smooth atom of supported near Q of Fβ(·,0),φp(·,0) (Rn−1) with

K ∈ ((s(·, 0))+ +max0, log2 c1,∞) , (4.12)

L ∈(

n− 1

min1, (p(·, 0))−− (n− 1)− (s(·, 0))−,∞

)(4.13)

and ∥∥∥∥λ(i)

Q

Q∈Q∗(Rn−1)

∥∥∥∥fβ(·,0),φp(·,0)

(Rn−1)

<∞, (4.14)

where, for all Q ∈ Q∗(Rn−1),

λ(i)

Q:= [ℓ(Q)]−

12 tQ×[(i−1)ℓ(Q),iℓ(Q)).

By (4.2), we see that

n− 1

min1, (p(·, 0))−− (n− 1)− (s(·, 0))− < 0

and then, by Remark 3.7(i), we know that the vanishing moment for (K, L)-

smooth atoms of Fβ(·,0),φp(·,0) (Rn−1) is avoid. Since supp aQ ⊂ 3Q, K ≤ K and, for

all α ∈ Zn+, with |α| ≤ K, and all x ∈ Rn, |DαaQ(x)| ≤ 2(|α|+n/2)j , it follows that,

for i ∈ 0, 1, 2, α ∈ Zn+, with |α| ≤ K, and all x ∈ Rn,

|Dαb(i)Q (x)| ≤ 2(|α|+n/2)j

and supp b(i)Q ⊂ 3Q. Thus, for i ∈ 0, 1, 2, b(i)

Qis a (K, L)-smooth atom sup-

ported near Q of Fβ(·,0),φp(·,0) (Rn−1) with (K, L) as in (4.12) and (4.13).


Let λ(i) := λ(i)QQ∈Q∗(Rn−1), where i ∈ 0, 1, 2. Next we show that, for any

given dyadic cube P ⊂ Rn−1,

1

φ(P )

∥∥∥∥∥∥∥∥∥

∞∑

j=(jP∨0)

∑

Q∈Q∗(Rn−1)

ℓ(Q)=2−j

[2j(β(·,0))|λ(i)

Q||Q|− 1

2χQ

]p(·,0)

1p(·,0)

∥∥∥∥∥∥∥∥∥Lp(·,0)(P )

is finite. By ‖f‖F

s(·),φp(·),p(·)

(Rn)= 1 and (4.11), we see that there exists a positive

constant C0 such that, for all P ∈ Q(Rn),

1

φ(P )

∥∥∥∥∥∥∥∥

∞∑

j=(jP∨0)

∑

Q∈Q∗(Rn)

ℓ(Q)=2−j

[2js(·)|tQ||Q|−

12χQ

]p(·)

1p(·)

∥∥∥∥∥∥∥∥Lp(·)(P )

≤ C0,

which, together with Remark 1.1(ii), implies that, for all P ∈ Q(Rn),

∫

Rn

∞∑

j=(jP∨0)

∑

Q∈Q∗(Rn)

ℓ(Q)=2−j

[2js(·)|tQ||Q|−

12

χPC0φ(P )

χQ

]p(·)dx ≤ 1. (4.15)

On the other hand, for all dyadic cube P ∈ Q(Rn−1), we have

I(P ) :=

∫

Rn−1

∞∑

j=(jP∨0)

∑

Q∈Q(Rn−1)

ℓ(Q)=2−j

2jβ(x,0)

|λ(i)Q|

|Q| 12χP (x)χQ(x)

C0φ(P )

p(x,0)

dx

=∞∑

j=(jP∨0)

∑

Q∈Q(Rn−1)

ℓ(Q)=2−j

2−j∫

Q

[2js(x,0)|λ(i)

Q||Q|− 1

2χP (x)

C0φ(P )

]p(x,0)dx

∼∞∑

j=(jP∨0)

∑

Q∈Q(Rn−1)

ℓ(Q)=2−j

∫

Qi

2js(x,0)

|λ(i)Q|

|Q| 12χP (x)

C0φ(P )

p(x,0)

dxdxn

.

∞∑

j=(jP∨0)

∫

2(P×[0,ℓ(P )))

∑

Q∈Q(Rn−1)

ℓ(Q)=2−j

[2js(x,0)|λ(i)

Q||Q|− 1

2

×[C0φ(P )

]−1

χ Qi

(x, xn)

]p(x,0)dxdxn,

where

Qi := Q×

[(2i− 1)ℓ(Q)

2, iℓ(Q)

),


which, combined with the fact that QiQ∈Q∗(Rn−1) are disjoint each other, (4.15)

and the condition (S1) of φ, implies that, for all P ∈ Q(Rn−1),

I(P ).

∫

2(P×[0,ℓ(P ))

∞∑

j=(jP∨0)

∑

Q∈Q(Rn−1)

ℓ(Q)=2−j

2js(x,0)|λ(i)Q||Q|− 1

2

× 1

C0φ(P )χQ×[

(2i−1)ℓ(Q)2

,iℓ(Q))

]p(x)

p(x,0)p(x)

dxdxn

.

∫

2(P×[0,ℓ(P ))

∞∑

j=(jP∨0)

∑

Q∈Q(Rn−1)

ℓ(Q)=2−j

2js(x)|tQ×[(i−1)ℓ(Q),iℓ(Q))|

× |ℓ(Q)|−n2

C0φ(P × [0, ℓ(P )))χQ×[ (2i−1)ℓ(Q)

2,iℓ(Q))

]p(x) dx . 1,

where we used the fact that p(x, 0) = p(x, xn) for all (x, xn) ∈ Rn with |xn| ≤ 2in the last inequality. By this and Remark 1.1(ii), we conclude that, for all

P ∈ Q(Rn−1),

1

φ(P )

∥∥∥∥∥∥∥∥∥∥

∞∑

j=(jP∨0)

∑

Q∈Q∗(Rn−1)

ℓ(Q)=2−j

2jβ(·,0)|λ(i)Q||Q|− 1

2χQ

p(·,0)

1p(·,0)

∥∥∥∥∥∥∥∥∥∥Lp(·,0)(P )

is finite, which implies that ‖λ(i)‖fβ(·,0),φ

p(·,0)(Rn−1)

. 1, namely, (4.14) holds true.

Therefore,

‖Tr(f)‖F

β(·,0),φp(·,0)

(Rn−1).

2∑

i=0

‖λ(i)‖fβ(·,0),φp(·,0)

(Rn−1). ‖f‖

Fs(·),φp(·),p(·)

(Rn).

Step 2) We prove that the operator Tr is surjective. Let f ∈ Fβ(·,0),φp(·,0) (Rn−1).

Then, by Theorem 3.8, we find that there exist a sequence

λ := λQQ∈Q∗(Rn−1) ⊂ C

and a sequence aQQ∈Q∗(Rn−1) of (K, L)-smooth atoms of Fβ(·,0),φp(·,0) (Rn−1) with K

and L satisfying (4.10) such that f =∑

Q∈Q∗(Rn−1) λQaQ converges in S ′(Rn−1)and

‖λ‖fβ(·,0),φp(·,0)

(Rn−1). ‖f‖

Fβ(·,0),φp(·,0)

(Rn−1); (4.16)


moreover, for all P ∈ Q(Rn−1),

1

φ(P )

∫

P

∞∑

j=(jP∨0)

∑

Q∈Q∗(Rn−1)

Q⊂P

ℓ(Q)=2−j

2jβ(x,0)|λQ|χQ(x)|Q| 12‖λ‖

fβ(·,0),φp(·,0)

(Rn−1)

p(x,0)

dx . 1. (4.17)

Let η ∈ C∞c (R) satisfy supp η ⊂ (−1

2, 12) and η(0) = 1. For all Q ∈ Q(Rn−1) and

ξ ∈ R, let ηQ(ξ) := η(2− log2 ℓ(Q)ξ),

g :=∑

Q∈Q∗(Rn−1)

λQaQ ⊗ ηQ =:∑

Q∈Q∗(Rn)

tQbQ,

where, for all Q ∈ Q∗(Rn) and x := (x, xn) ∈ Rn,

bQ(x) := [ℓ(Q)]−12aQ ⊗ ηQ(x) := [ℓ(Q)]−

12aQ(x)ηQ(xn),

tQ := [ℓ(Q)]1/2λQ if Q = Q× [0, ℓ(Q)) and tQ := 0 otherwise.

Next we show that g converges in S ′(Rn) and

‖g‖F

s(·),φp(·),p(·)

(Rn). ‖f‖

Fβ(·,0),φ

p(·,0)(Rn−1)

.

It is easy to show that each bQ is a (K, L)-smooth atom supported near Q of

Fs(·),φp(·),p(·)(R

n) with K and L as in (4.10). By Proposition 4.6 and the fact that

Q× [12ℓ(Q), ℓ(Q))Q∈Q∗(Rn−1) are disjoint each other, we find that

‖tQQ∈Q∗(Rn)‖fs(·),φp(·),p(·)

(Rn)

= supP∈Q(Rn)

1

φ(P )

∥∥∥∥∥∥

∞∑

j=(jP∨0)

∑

Q∈Θj

2js(·)|λQ|[ℓ(Q)]1/2

× |Q× [0, ℓ(Q))|−1/2χQ×[0,ℓ(Q))

]p(·)

1/p(·)∥∥∥∥∥∥∥Lp(·)(P )

∼ supP∈Q(Rn)

1

φ(P )

∥∥∥∥∥∥

∞∑

j=(jP∨0)

∑

Q∈Θj

2js(·)|λQ||Q|−12χQ×[ 1

2ℓ(Q),ℓ(Q))

∥∥∥∥∥∥Lp(·)(P )

∼ supP∈Q(Rn−1)

1

φ(P )

∥∥∥∥∥∥∥∥

∑

Q∈Q∗(Rn−1)

Q⊂P

|Q|−s(·)n−1 |λQ|

×|Q|−1/2χQ×[ 12ℓ(Q),ℓ(Q))

∥∥∥∥∥Lp(·)(P×[0,ℓ(P )))

, (4.18)


where Θj := Q ∈ Q∗(Rn−1) : Q × [0, ℓ(Q)) ⊂ P, ℓ(Q) = 2−j. On the other

hand, let Γ := ‖λ‖fβ(·,0),φp(·,0)

(Rn−1). Then, for all P ∈ Q(Rn−1), by (4.17), we find

that

1

φ(P )

∫

P×[0,ℓ(P ))

∑

Q∈Q∗(Rn−1)

Q⊂P

|λQ||Q|−12Γ−1|Q|−

s(x)n−1χQ×[ 1

2ℓ(Q),ℓ(Q))

p(x)

dx

∼ 1

φ(P )

∑

Q∈Q∗(Rn−1)

Q⊂P

∫

Q×[ 12ℓ(Q),ℓ(Q))

[|Q|−

s(x)n−1

− 12 |λQ|Γ−1

]p(x)dx

∼ 1

φ(P )

∞∑

j=(jP∨0)

∑

Q∈Q∗(Rn−1)

Q⊂P

ℓ(Q)=2−j

∫

Q

[2jβ(x,0)|λQ||Q|−

12Γ−1

]p(x,0)dx . 1,

which, together with Remark 1.1(ii), implies that

1

φ(P )

∥∥∥∥∥∥∥∥

∑

Q∈Q∗(Rn−1)

Q⊂P

|Q|−s(·)n−1 |λQ||Q|−

12χQ×[ 1

2ℓ(Q),ℓ(Q))

∥∥∥∥∥∥∥∥Lp(·)(P×[0,ℓ(P )))

. ‖λ‖fβ(·,0),φ

p(·,0)(Rn−1)

.

Form this and (4.18), we further deduce that∥∥∥tQQ∈Q∗(Rn)

∥∥∥fs(·),φp(·),p(·)

(Rn). ‖λ‖

fβ(·,0),φ

p(·,0)(Rn−1)

.

Therefore, by Theorem 3.8(i) and (4.16), we conclude that g =∑

Q∈Q∗(Rn) tQbQ

converges in S ′(Rn), g ∈ Fs(·),φp(·),p(·)(R

n) and

‖g‖F

s(·),φp(·),p(·)

(Rn). ‖f‖

Fβ(·,0),φ

p(·,0)(Rn−1)

;

furthermore, Tr(g) = f in S ′(Rn−1), which implies that

Tr : Fs(·),φp(·),p(·)(R

n) → Fs(·,0),φp(·,0),p(·,0)(R

n−1)

is surjective and hence completes the proof of Theorem 4.1.

Acknowledgement. The authors would like to express their deep thanks toreferees for their careful reading and many useful comments which improve thepresentation of this article. This project is supported by the National Natu-ral Science Foundation of China (Grant Nos. 11171027, 11361020 & 11471042),the Specialized Research Fund for the Doctoral Program of Higher Education ofChina (Grant No. 20120003110003) and the Fundamental Research Funds forCentral Universities of China (Grant Nos. 2012LYB26, 2012CXQT09, 2013YB60and 2014KJJCA10).


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1 School of Mathematical Sciences, Beijing Normal University, Laboratory

of Mathematics and Complex Systems, Ministry of Education, Beijing 100875,

People’s Republic of China.

E-mail address : [email protected] address : [email protected] address : [email protected]

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